CONTRIBUTIONS TO THE THEORY
OF RESTRICTED POLYNOMIAL
AND RATIONAL APPROXIMATION

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
KATHLEEN ANN TAYLOR
1970

THEGIS

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This is to certify that the

thesis entitled

Contributions to the Theory of
Restricted Polynomial and Rational

Approximation
presented by
Kathleen Ann Taylor

has been accepted towards fulfillment

of the requirements for

_Ph_'.D_°_ degree in wt iC S

MDM

Major professord

Dm 6-26-70

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ABSTRACT
CONTRIBUTIONS TO THE THEORY

OF RESTRICTED
POLYNOMIAL AND RATIONAL APPROXIMATION

BY

Kathleen Ann Taylor

In Chapter I we consider the problem of approximating
functions continuous on a compact metric Space S by elements
of a linear subspace V of C(S) in the following manner:

1. J, K, and L are compact subsets of S.

2. Two prescribed functions L and p are given
and are assumed to be continuous on L and J respectively.

3. We allow as approximants the subset V1 of V
whose elements v are such that v(x) S u(x) for all x E J
and v(x) 2 L(x) for all x E L.

4. For a given f E C(S), a best approximation

v0 6 V1 will be such that

max |v (x) - £(x)| = min {max \v(x) - £(x)|}.
x€K o VEV1 x€K

The existence of best approximations follows from
the usual compactness arguments. For functions f 6 C(S)
such that {(x) 5 f(x) for all x E L and p(x) 2 f(x)
for all x E J, best approximations can be characterized
in terms of a linear functional based on the set of critical

points. There is a unique best approximation for each such

Kathleen Ann Taylor

f if and only if V is a Haar Sub8pace. A Remes-type
algorithm is given to construct such best approximations.

We let V be a set of rational functions in Chapter
II and consider the same problem. If we properly restrict
the functions L and p and the sets L and J we obtain
existence theorems, and for suitable f's we again characterize
best approximations in terms of a linear functional based on
the set of critical points. In Special situations we have
uniqueness of the best approximation.

An expository presentation of the doctoral thesis of
Karl-Heinz Hoffmann is included (Chapter III) because it pre-
sents a very general theory of restricted approximations.

The relationship of his work to the results presented here is

discussed.

CONTRIBUTIONS TO.THE THEORY
OF RESTRICTED
POLYNOMIAL AND RATIONAL APPROXIMATION

BY

Kathleen Ann Taylor

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHIIDSOPHY

Department of Mathematics

1970

fin. ;.

 

.2}. (1.5 m9
/-- a7- "7/

ACKNOWIEDGEMENTS

I wish to express my gratitude to my major Professor
Gerald D. Taylor for his help in the preparation of this
paper. His many suggestions and generous donation of time
were of immeasurable assistance. I also wish to thank
Professor Mary J. Winter for her encouragement and faith in

my ability.

ii

Chapter

I

II

III

TABLE OF CONTENTS

INTRODUCTION ...... ......... ...................

CHEBYSHEV APPROXIMATION WITH GENERALIZED
POLYNOMIALS HAVING RESTRICTED RANGES:

INMUALITY CASE ..... O O O O O O O O O O O O O C O I O I O O O O O O O 0

Section 1: Basic Definitions and Existence
Theorem 0 O O C O Q C C I O O O O O ........ O O O O 0

Section 2: Kolmogorov-type Characterization
Theorem 0 O O C O ...... O O O O O I O C I O O O O O O 0

Section 3: Uniqueness and Related Results ....
Section 4: Remes Algorithm for Calculating
Best Restricted Approximation .....

CHEBYSHEV APPROXIMATION WITH RATIONAL FUNCTIONS
HAVING RESTRICTED RANGES: CONSIDERATION OF
mUALITY IN THE BOUNDS ooooeoooooooeoooooooooeo

Section 1: Introduction ......................
Section 2: The Existence Problem .............
Section 3: Characterization of Best Restricted

Rational Approximations ...........
Section 4: Equality in the Bounding Curves ...
Comments: ...................................

NON‘IINEAR CHEBYSHEV APPROXIMATION WITH SIDE
CONDITIONS 00000000000000.000000coo-o.oooocoooo

Section 1: Definitions and Statement of the
Problem and Standard Theory .......
Section 2: Structure of V and Properties
of the Side Conditions ............
Section 3: A Special Class of Non-linear
Approximation Problems ............

Section 4: Results Concerning Uniqueness .....
Seetion 5: A speCial Case ......OOOOOOOOOOOOOO
Section 6: Relation of Chapter III to Chapters

Iand II .....OOOOOOOOOOOOOOO 000000

BIBLIOGRAHIY OOOOOOOOOOOOOOOOOOOOOOO0.0....0...

iii

55

55
56

63

86
101

103

103

109

126
136

142
143

154

INTRODUCTION

This paper will consider the approximation of func-
tions in a normed linear space by functions from some subset
of that space. Let C(S) be the linear space of continuous
real-valued functions on a space S normed with the uniform
norm. Let V be a subspace of C(S) and V1 a subset of
V whose elements satisfy certain prescribed conditions. We
shall examine questions of existence, characterization,
uniqueness, and computation of a best uniform approximation
to a given function f in C(S) by elements of V1-

The problems considered in Chapters I and II are a
combined generalization of work done by P.J. Laurent [13]
and G.D. Taylor [21], [22], [23], [24].

In the paper by Laurent, S is the union of two compact
Spaces K and L and V is a finite dimensional subspace of
C(S). For a fixed f E c(S), V1 is the subset of V whose
elements are less than or equal to f on L. In this setting,
Isurent considers the problem of approximating f by elements
of V1 where the error in the approximation of f by v 6 V1
is defined to be the maximum of ‘f(x) - v(x)‘ on K.

In the work by Taylor, S is a compact subset of the

real line and V is a finite dimensional Haar subSpace (the

definition of a Haar subspace will be given later) of C(S).

For two fixed extended real-valued functions L and u
defined on S, V1 is the subset of V whose elements are
less than or equal to p and greater than or equal to L
on S. The error in approximating a given f E C(S) by
v 6 V1 is defined to be the maximum of If(x) - v(x)I on S.

This paper will consider S to be the union of three
compact Spaces J, K, and L, and V a finite dimensional
subSpace of C(S); L and u will be fixed real-valued
functions with L continuous on L and p continuous on
J. V1 will consist of elements of V which are greater
than or equal to L on L and less than or equal to u on
J. For a function f E C(S), the error in approximation by
v 6 V1 will be defined to be maximum ‘f(x) - v(x)\ on K.
In Chapter I we shall assume L(x) < p(x) for all
x E J n L; in Chapter II we shall consider what happens when
L(xj) B‘iij) for j = l,...,n. The set V may be gen-
eralized polynomials or rational functions.

Chapter III is an expository presentation of the work
of K.H. Hoffmann [7] on non-linear Chebyshev approximation
with side conditions. Some remarks are made concerning the
application of Hoffmann's work to the problems considered in
Chapters I and II. Also some additional comments are made

concerning his uniqueness results.

CHAPTER I
CHEBYSHEV APPROXIMATION WITH GENERALIZED

POIXNOMIALS HAVING RESTRICTED RANGES:
INEQUALITY CASE

Section 1: Basic Definitions and Existence Theorem.

Let J, K, L be three (not necessarily disjoint)
compact subsets of a metric space and let S = J U K U L,
and assume K contains at least n points. By C(S) we
Shall mean the space of continuous real-valued functions f
with the topology induced by the Chebyshev or uniform norm

\Iwa = max |£(x)\.
x68

We Shall denote by H-HK the seminorm on S as follows:
for f E C(S)
HfHK = max ‘f(x)\.
XEK
Let w1,...,wn be linearly independent elements of

C(S) and let V be the subSpace of C(S) generated by

w ,...,w . Let L and be real-valued functions continuous

l n p

on L and J respectively. In this chapter, we shall assume
LCX) < u(x) for all x 6 J n L.

Let V1 be the Subset of V consisting of those

elements bounded above by u at each point of J and below

by L at each point of L, i.e.

VI = {v E V: v(x) s u(x) for all x E J and

v(x) 2 L(x) for all x E L}.

We Shall assume V is non-empty. For a given real-

1
valued function f on S, we wish to find an element of V1
which is closest to f in the sense of minimizing the semi-
*
norm H-HK. That is, we want v 6 V1 such that

If - Jul. = i2; IIf - vuK 9.
V 1

If such a v* exists, it will be called a best restricted
approximation to f on K.

In the work done by G.D. Taylor [23] concerning approx-
imation by functions with restricted ranges on a compact sub-
set X of the real interval [a,b], the functions L and u
were assumed to be extended real-valued functions with the
following restrictions:

(i) L may assume the value -m, but not 1+».

(ii) u may assume the value +m, but not -m.

(iii) X_co = {x: L(x) = -m} and X+00 = {x: u(x) = +m}
are open subsets of X.
(iv) L is continuous on X ~ X_OD and p is continuous
on X ~ X+°.
In the present setting, no generality is lost by assuming that
-L and p are continuous on L and J reSpectively. Indeed

let L and p be extended real—valued functions defined on L

and J, reapectively, satisfying the above conditions; define

of,

isc

L,e_

L' = L ~ X and J' = J ~ x+w. Then L and U are con-

-a)

tinuous on the compact sets L' and J'. It is clear that

the subset V corresponding to L and p on L and J

l

is the same as Vi corresponding to L and u on L' and

J'. For convenience of notation we let L = L' and J = J'.

Theorem 1.1: (Existence) Let f 6 C(S) be given. Then a

best restricted approximation to f on K exists.

1 2 p. If w is any

element of V1 Such that “V - WUK > 2p1, then

Proof: Let v 6 V1. Then Hf - VHK = p

\If - WIIK 2 \Iv - qu - IIf - vuK > .1 2 p.

Therefore we need only consider approximation by elements of

the set B where
B = {w 6 V1: “v - wHK s 2p1}.

That is, p = inf Hf - wHK. But B is a closed, bounded sub-
WEB

set of a finite dimensional normed linear Space and therefore

compact. Since the seminorm H.HK is a continuous function

on B, the infimum is attained and a best restricted approx-

imation exists. II

Section 2: Kolmogorov-type Characterization Theorem.

In the classical problem of Chebyshev approximation
of a continuous function f on a compact set X by elements
*
of a linear subSpace V of C(X), the best approximation v

is characterized by properties of the set of extreme points,

i.e. the set

E = {x 6 X: [f(x) - v*(x)I = Hf - V*Hm}.

In 1948, Kolmogorov [12] proved that v* 6 V was a best
approximation to f if and only if

min (f(x) - v*(x))v(x) s o

xEE
for all v 6 V. By altering the set E, G.D. Taylor [ZI]
characterized the best approximation to f in the case of
restricted approximation on a compact set. And P.J. Laurent
[I3 was also able to characterize one-sided approximation on
two compact sets by this property. We Show that similar
modifications in the set E make possible a characterization
of the best restricted approximation considered here.

For any v E V define the function eV 6 C(S) by

1’

ev(x) = f(x) - v(x) for all x E S.

Now define the following sets of critical points for

E. = Ix 6 K ev<x> = IIevIIK}.
E" -- {x e K ev(x) =‘IIevIIKI°
G: = {x E L v(x) - L(x)}.
c; = Tx 6 J: v(x) = u(x)}.

_ + ' _ + - ' =
Let Ev - Ev U Ev and Gv - Gv U Gv' USing Dv Ev U Gv

88 our new set of "critical points", we can obtain the follow-

ing Kolmogorov-type theorem.

"'7

 

* *
Theorem 1.2: Let f e C(S) ~ V1 and v 6 V1. Then v is
not a best restricted approximation to f if there exists a

function v E V such that

+
v(x) > o for all x e E * u c+*
v

V

and

v(x) < O for all x E E-* U G *.
v v

Now suppose there exists a v0 6 V with

vo(x) > L(x) for all x E L
and

vo(x) < u(x) for all x E J,

then the converse is also true.
Proof: If such a v exists, we shall Show that there is a
*
positive number 6 such that v +-6v is a better restricted

approximation to f. Let Hvu0° = M. For x e E *,
V

\f(x) - v*(x)\ = Hf - v*HK.

.. IIf - v*uK

Let 6 and define the sets
0 2M

*

* Hf—v IIK

01 = {x G S: f(x) - v (x) > ———§—- and v(x) > O},

*

f-
II v I,

*
02 = [x 6 S: f(x) - v (x) < - and v(x) < 0}.

2

For 5550 and yEOI.

* ~k *
o s f(y) - (v + av) (y) = My) - v (y) - 6V(y) < IIf-v IIK-

Similarly, for 6 S 60 and y 6 O2

’Uf - v*HK < f(y) - v*(y) - 6v(y) = f(y) - (v* + 6v)(y) ; 0.

Now 0 = 01 U 02 is an open set and E * c 0. Thus K ~ 0
v

is compact and there is a number 31 > 0 such that for

YEK~03

If(y> - v*<y>\ s Hf - v*nK - e1.

6

If we choose 5 = 5%? then for 6 s 5 and y 6 K ~ 0

l 1

€
Ho>-$u>-monsnf-$M-e +4<uf-$

1 2 HK'

61
So for 6 s Efi’,

Hf—W*+MmfflH-vmr

It remains to be shown that 5 can be chosen so that
x
V +'6v 6 V1.
+ +
For x 6 G *, v(x) > 0. By the compactness of G *,
v v

. . . +
there IS an open set U containing G * such that for y E U
v

V(y) > 0.

and

v*<y> +-6v(y> 2 L(Y) +~av(y> > L(Y)-

Since L ~ U is again compact, there is a number > 0

82
such that for y E L ~ U,

v*<y> 2 L(y) +‘ez-

(-2

Then for 6 < 2%. and y E L ~ U,

* 62
V (y) + 6V(y) 2 MY) +T>t(y)o

6

Thus for y E L, 6 < 53' implies

(v* + 6v)(y) > My)-

For x E G-*, v(x) < 0. By the compactness of G-*,
v v
there is an open set W containing G-* such that for y E W,
v

v(Y) < 0:

and

v*<y> + My) s My) + (NO) < My)-

Since J ~ W is again compact, there is a number > 0

63
Such that for y E J L W,

*
v (y) smy) - .3.

G

SOfOI' O<'2_%and YEJFW:

* 63
V (y) + 6V(y) SMy) - 7< My)-

By choosing 6 such that
6 < min {31,62,63}/2M9

*
we obtain v +'6v 6 V1, a better restricted approximation
to f.
I * I
Conversely, if v is not the best restricted approx-

imation, let w be a better approximation, i.e.

10

If - A. < If - $11K-

Then for x E E *,
v

|f(x) - v*(x)I = Hf - v*\\K > \If - qu 2 Iraq w(x)|.

Thus

sgn (w(x) - v*<x)> = sgn [f(x) - v*<x) - (f(x) mm
= sgn [f(x) - v*<x)1

*
and Iw(x) - v (x)\ 2 d > 0 for some d and all x 6 E *.

Consider
w*<x> = Ifig-(w<x) +-svo(x>>.

Let “V* - v0“co = M. For M ¢ 0 and 0 < 6 < gfi3

w*(x) - v*(x> = -1- (w(x) - v*(x)) + f3;- (vow v*(x>)

1+6

is such that

x
w (x) - v*(x) > 0 for x G E+;
v

and

x x -
w (x) - v (x) < O for x E E *.
v

For x 6 G+;, V*(x) = L(x). So

v

w(x) - v*(x) 2 O and vo(x) - v*(x) > 0.
Thus w*(x) - v*(x) > O.

- *
For x 6 G *, v (x) = u(x). So
v

11
x *
w(x) - v (x) s O and vo(x) - v (x) < O.

* x * *
Thus w (x) - v (x) < O, and w (x) - v (x) E V is the

desired function v.

* + -
If M = “v - von = 0, G * = G * = ¢ and we have
v
shown

* +
w(x) - v (x) > O for x 6 E *
v

w(x) - v*(x) < 0 for x E E-*.
v

*
So the function w(x) - v (x) E V satisfies the theorem. II

Remark 1: The extra hypothesis on V required for the con-
verse of this theorem will be discussed in the next section.

+ + - - # o 1
Remark 2: If (E * U G *) n (E * U G *) ¢ for a particu ar

*v v v v
f E C(S) and v E V1, then the v of Theorem 1.2 cannot

*

exist and v must be a best restricted approximation to f
even if the extra hypothesis is not satisfied by V. The con-
clusions drawn here are the same as those drawn by G.D. Taylor
[23 and are included here for completeness.

+ - *
1. If E * n E * ¢ ¢, then f - v E O on K. This

v v
can occur even if f i v on S.

2. 1f E+; n G-* # ¢, then for some x E K.n J,
v v

'k *
f(x) - v (x) = “f - v “K

and

*
v (X) = IJ-(X)-

12

To get closer to f at this point, we would have to have
*
vo(x) > u(x) thus removing vO from V1. So v must be

a best restricted approximation.

3. If E-* n G+; # ¢, then for some x E KI] L,
v v

f(x) - v*(x) = 'Hf - v*HK
and

'k
X (X) = UK)-

Again, to choose v0 6 V closer to f, vo(x) < L(x) which
*

would mean vo E V1. Thus v is a best approximation.

4. Let C(S) = {f E C(S): f(x) 2 L(x) for all

x E L, f(x) s.p(x) for all x E J, and p

inf f - v 0 .
vele HK > I

Then for f E C(S) and any v 6 V1,

+ + - -
(Ev U Gv) n (Ev U Gv) 8 ¢.

Section 3: Uniqueness and Related Results.

In this section we shall use Theorem 1.2 to obtain
characterization and alternation theorems which will be
important in constructing an algorithm to determine the best
restricted approximation. Laurent, in his one-sided approx-
imation problem, assumed that the set V1 contained an element
strictly less than the f to be approximated. This enabled
him to characterize the best approximation by means of a linear
functional on C(S) based on the critical points of the error
function with at least one of these points a maximum point for

the absolute value of the error curve. We shall make a similar

13

assumption here in condition H and Show that this assumption

is satisfied in the Special case that V is a Haar subSpace.

Condition H: We Shall say V satisfies condition H provided

there is a v 6 V1 Such that

v(x) < u(x) for all x € J
and

v(x) > L(x) for all x E L.

We can now characterize the best restricted approx-

*
imation v to f from 4V1 when V1 satisfies condition

H and (E+; U G+;) n (E-* U G-*) = ¢ by means of a continuous
v v v v
linear functional L in C(S) whose null Space contains V.

Theorem 1.3: Let V satisfy condition H. Then a necessary

*
and sufficient condition for v E V1 to be a best restricted

approximation to f E C(S) is that there exist k (s n+1)

critical points

x1,x2,...,xk in Dv*

such that {x1,x2,...,xk} 0 Ev* * ¢, and a linear functional
L defined by

k
L(h) = z: lih(xi).

i=1

such that L vanishes on V and

uc;+

+
*1 > O for X1 G E v*,

*
V

Ii < 0 for X1 E E * U G *.
V V

14

*
Proof: (Sufficiency) Suppose v satisfies the hypotheses
and that w E V1 is a better restricted approximation to f.

Then

If - A, < IIf - v"‘IIK .

*
and v = w - v E V is such that

+
v(xi) > 0 for x E E * (Ii > O),

v(xi) < 0 for x E E * (Ii < 0).

V

v(xi) s o for xi 6 c', (ii < 0),
V
+

v(xi) 2 0 for X1 E Gv* (xi > 0).

Since at least one xi E E * by hypothesis, L(v) > O. This
v

is a contradiction to L vanishing on V, so v* is a best
restricted approximation.
(Necessity) Let {w1,...,wn} be a basis for V.
Let v* E V1 be a best restricted approximation to f with
correSponding set D *. Denote by F the S€t
v

+;}

n _ + .
F = {(2 ..,zn) E R : zi - wi(x) for x E E * U G

V V

1"

n . _ _ - - .
U {(zl,...,zn) E R . zi wi(x) for x E Ev* U Gv*}

If 6 e R“ is not in co (r) (the convex hull of r), by

the theorem on linear inequalities [3 , p. 19], there exists
n
. a point (c ,...,c ) e R“ such that z c z. > o for all
1 n i=1 i 1
(21,...,zn) E P. But then the function v E V defined by
n
2 c w (x) I v(x) is Such that
i=1 1 i

15
v(x) > 0 for x E E+; U G+;,
v
v(x) < 0 for x E E-* U G *.

x
This contradicts the fact that v is a best restricted
approximation (see Theorem 1.2). So 6 E Rn must be in
co (P). Then by the theorem of Caratheodory [3 , p. 17] we

can find k (s n+1) points z ...,z in F, and k positive

1’ k

real numbers a1,...,ak, such that

k
O= 233.2.
i=111
and
k
1 = E a .
i=1 i

Since zi = ixw1(xi),w2(xi),...,wn(xi)) ‘we have

R
O 3 .2 aieiwj(xi) for j 8 1,...,n,
i=1
with
, + +
+11fxiEE*UG*,
v v
e:

v v
Set *1 = aiei and
k
= h ,
MM 121% (xi)

L is a linear functional on C(S) vanishing on V. We

must Show that at least one of the xi's is in E *. Suppose
v
not: then for each v E V we have that

16

+
v(xi) L(xi) for all xi E Gv*

and

v(xi) = ”(xi) for all xi 6 c‘*.
V

+
Indeed, if there is xi E G * for which
v

my > L(xi) = v*<xi).

then (v(xi) - v*(xi))),i > O

*
implying L(v - v ) > O which is a contradiction. So at
least one xi must be in E.*. II
v

The proof of the sufficiency did not require that

(E+; U G+;) n (E-* U G-*) = ¢. This condition is required
v v v v

for the proof of the necessity as shown by the following

example:

Example 1.1: Let f(x) = 1 - x2 on the real interval [-1,1].
We wish to approximate f by polynomials of degree at most two
which are less than or equal to 0 = u(x) and greater than or

equal to -1 - L(x) on [-l,l].

 

 

-‘ I X

 

 

B |

3""

17

Then one best restricted approximation to f is

*
v (x) = - %’X2.

+ -
The correSponding set D * = {O} = E * = G A non-zero
v * v v

. . 2 .
functional L cannot exist for v Since {1, x, x } is

*0

a basis for V and if L(h) = xh(0) is to vanish on V, we

must have
L(l) = I = 0.

An n-dimensional subSpace V c:C(S) is called a Haar

subSpace if every non-zero element of V has at most n-l zeros.

Remark 1: In the previous theorem, if V is a Haar subspace,
k

then k = n+1. Since if k,< n+1 and Z kiwj
i=1

(Xi) = O for

S all different from x1,...,xk and setting the correSponding
I = 0.

fl

.2 xiwj(xi) = O for j = 1,...,n,
i=1
That is, det (wj(xi)) = 0 so there exist real numbers

Bl”"’8n’ not all zero, such that

n
2 Biwi(x_1) = 0 for J = 1,...,n.
i=1
n
But then the function v(x) = 2 Biwi(x) E‘V has n zeros,
i=1

which contradicts the Haar condition since v i 0. So

k = n+1.

The
is

the

hUs

18

Remark 2: If S is a compact subset of the real line, V is

a Haar subspace of dimension n on a closed interval [a,b]
properly containing S, and V1 contains at least two distinct
elements, then V satisfies condition H.

M: Let v1(x) and v2(x) E V1 with v1 i v2. If either
v1 or v2 satisfies condition H, we are done. Suppose not.
Let v0 8 (v1 +v2)/2, vo E V1. If x E G; = {x E J: vo(x)

o
p. (30}, then

v1(x) v2(x) = p(x).

+
Similarly, if x E Gv = {x E L: vo(x) B L(x)}, then
0

v1<x> = vzm = Lot).

So vo mst meet p, and L at most n-l times since V
is a Haar subSpace and v1 - v2 can have at most n-l zeros.

If J n L - ¢, construct v E V such that for some 6 > 0,

+6 for x E G: ,
o
v(x) I

-6 for x E G- .
"0

Then there is an Open set U containing G: on which v(x)
o
is positive. L ... U compact implies there is an 31 > 0 such

that

vo(x) - L(x) 2 e for all x E L .. U.

l

C
1
Thus for 'I]<-“—“-,
2v
Q

v00!) + ‘nv(x) > L(x) for all x E L.

19

Also there is an open set W containing G; on which v(x)
o
is negative. J ~ W compact implies there is an QZ > 0 such

that

u(x) ' V0(X) 2 62 for all x E J ~ W.

62

Thus for n > EW;U—'a
a)

vo(x) + “v(x) < u(x) for all x E J.

Then “‘< min {€1’€Z}/2Hvum’ implies vo(x) + nv(x) E V1

and does not intersect either L(x) or u(x).

If J n L # ¢, order the points in G:' U CV and

o o
label them

Without loss of generality, assume x1 E 63'. Group these

0
points so that
x < < x G+ ' ts
1 ... k1 are v pOin ,
o
x <...< x are G- points,
k1+1 k2 v0
(-1)“‘
m m+l o

The finite interval [a,b] properly contains 8. Let

Y0 = a G< X1). Choose y1 such that

xk<y

<X
1 1 k1+l

20

Similarly choose yi, i = 2,...,m, and let = b (> xn_1).

ym+1
Construct v E V [ 8, p. 28] Such that

II
C
m
0
"1
H.

II
H

U
U
a

V(yi)

N
O

v(x) on [yo.y1],

v(x) s O on [y1,y2],

<-1>my(x)

N
O

on [ymwmfl] .

and v(x) # 0 for x E (a,b) N {y1,...,ym}.

Let 61 = min (MK) - V (X)).
o
xE[a,y1]nJ
52 = min (vo(x> - L(X)).
xEIy1.y2]nL
min (”(x) - vo(x)) if m is even,
XEIVm.b]nJ
5m+1 '3
min (v0(x) - L(x)) if m is odd.
xE[ym,b}]L
Let 6 = min 6,. 6 > 0 since each 6, is positive
i=l,...,m+1 l

by construction. Multiply the function v described above

by an appropriate positive number n such that

IIIIVIIQ < 5/2.
Then vo(x) + “v(x) E V1 does not intersect either L(x) or
u(x). Hence V1 satisfies condition H. II

The Haar condition on V also assures uniqueness of

the best approximation. The following theorem is analogous to

"u':

21

the Haar uniqueness theorem [3 , p. 81] in the Standard
Chebyshev approximation theorem. Let 01(8) = {f E C(S):

+ +» - - _
(Ev U Gv) 0 (EV U Gv) - ¢ for all v E V1}.

Theorem 1.4: Assume condition H is satisfied for V an n-

dimensional subspace of C(S). Then a necessary and sufficient
*

condition for a best restricted approximation v to f 6 01(8)

to be unique is that there does not exist a linear functional

k
L(h) = E Iih(xi)
i=1

on C(S) such that k s n, L vanishes on V, and at least

one xi E E *.
v

Egggfz (Sufficiency) Suppose no such functional L exists
and f has two best restricted approximations v1, v2. Then
v0 3 (v1 +-v2)/2 is also a best approximation and its char-
acterizing functional must be based on n+1 points. Let
these be x1,...,xn+1, and the functional
n+1
L(h) = 121 aih(xi).

The xi's must be critical points for v1 and v2 and

v(xj) = O for j = 1,...,n+1.

\IU

3:“

22

By assumption, one of the x '8, say xjo’ is an element of

J

E . Then let
vo

D = det {wi(xj): i = 1,...,n; j = 1,...,n+1; j # jo}.

Since L(wi) = 0 for i = 1,...,n,

n+1

321 “j“i‘xj) = '“Jo“i"‘

Ji‘JO

jo"

n+1
{QJ}J=1.J*JO

fixed, we can find another solution {a

If D = O, the solution is not unique. So for

, n+1
JO J j=1,j#j0’

051 = 0 for some jl' But this gives a functional L' such

a with

that
' n+1 '
L (h) = 121 aj'h(xj). (o o = ovjo)
m,
which vanishes on V. This is a contradiction and we conclude

D # 0. Since

n
2 B-W.(X ) = 0:
i=1 111

we must have Bi = 0 for i = 1,...,n; so v1 E v2.

(Necessity) Suppose condition H is satisfied and there

is a non-zero linear functional L based on n points x1,...,xn

which vanishes on V. Then

n

L(h) = 121 moi)

and the {xi}:=1 are a non-identically zero solution of

n

r—‘\

.L xiwj(xi) = O for j = 1,...,n
i=1

23
which implies det (wj(xi)) = O, and we can obtain a function

n
v(x) = 2 o wj(x) i 0
j=1 j

and

IIVIL, = 1’

such that v(xi) = 0 for i = 1,...,n. By condition H, there

is a function vO E V such that

l
vo(x) > L(x) for all x E L,

vo(x) < u(x) for all x E J.

Let 26 = min {min (vo(x) - L(x)), min (”(x) - vo(x))}. Let
xEL ~ xEJ
6' = min {l,6}.. Choose f(x) E C(S) such that

HH,=E@QI=V>0
and

~ *
sgn f(xi) = sgn xi-

* - sgn xi if *1 ¥ 0
(sgn I. -
1 +1 if I. =0 .

Set f(x) = f(x) (1 - Iv(x)l). Consider
F(x) = vo(x) + f(x).

Since If(x)I s 6', F(x) 2 L(x) for all x E L and
F(x) 5 u(x) for all x E J.

We claim: for any w E V1,

HF - w“0° 2 6'.

24

If w E V1 is such that
“F " WHQ< 6',

* x
then w = vo +-v with v E V so

*
F - w = f - v ,

and If(xi)I = 6'.

x * ‘
Thus sgn v (xi) = sgn f(xi) = sgn *1: i = 1,...,n. But
n
*
then 2 xiv (xi) > O which is a contradiction. Now consider
i=1

vo(x) + Xv(x).

If 0 s i s 5', v0 +-iy 6 v1. Moreover,

6'

V\

1%) - (v() + ma)!

\f(X) - AV(X)I

IA

I¥<x>I<1 - Iv<x)I) + i\v(x>I

IA

a'-(1 - Iv<x>I> + i\v(x)I

5' - (6' - i)\v(x)\ s 5' for o s i s 5'.

Thus, if we choose any I such that O S l S 6', the function
vo +>Xv is a best restricted approximation to F and we have
constructed a function whose best restricted approximation is
not unique. II

For the case that J = K = L, G.D. Taylor [23] proved
that if V is a Haar Subspace, then beat approximations are
unique for each f E C(S) which lies between the bounds. The

following corollary characterizes those subSpaces which yield

25
best approximations for all f E C1(S).

Corollary: ASSume V satisfies condition H. Then each

f 6 01(3), has a unique best approximation from ‘Vl if and
only if V is a Haar subSpace.

2:29;; (Sufficiency) Suppose f E 01(3) has two best
restricted approximations. Then, by Theorem 1.4, there is a
continuous linear functional L vanishing on V based on

k s n points. But by Remark 1 following Theorem 1.3, this
contradicts the Haar condition. Thus each f has a unique
best restricted approximation.

(If f E V , then f is the unique best restricted

1
approximation to f from V1 and it is unique since K
contains at least n points.)

(Necessity) If V is not a Haar subspace, there

exists a function v E V,

IIVII... = 1’

with distinct points x1,...,xn in S for which

v(x ) = 0 j = 1,...,n.

J

Then, as in the proof of Theorem 1.4, we can construct a
function F E C(S) which does not have a Unique best

restricted approximation. II

Alternation theorems are very useful in constructing and

recognizing the best approximation in the standard Chebyshev

 

lr-n'l

'I'E

[3’

th

11+

hit

26

approximation of functions by elements of a linear Space V.
G.D. Taylor [23] was able to Show that an alternation theorem
is also valid in ordinary restricted approximation if we
modify slightly the idea of alternations. A similar theorem
is valid in the problem presented here. For the proof of
this theorem, it will again be necessary to assume that

(E: U GI) n (E; U 6;) = (I) for the given function f and all

v EVI-

Theorem 1.5: Let S be a compact Subset of [a,b], V be
an n-dimensional Haar subSpace of C[a,b], and f E 01(8).

Then v E V is a best restricted approximation to f if

1

and only if there exist n+1 consecutive points

X (X <...<X

1 2 from Ev U Gv ,

n+1

with at least one xi E Ev’ such that for

+1 for x. €E+UG+a
i v v

0(Xi) _ _
-l for x. E E U G ,
i v v

we have moi) = <-1>”le<x1>.

Proof: (Necessity) If v is a best restricted approximation,

 

then Remark 1 following Theorem 1.3 implies that there exist

n+1 crit ical points

x <x ... x
1 2< < n’

with at least one in Ev’ and a linear functional,

27
n+1

L(h) = Z Lh(X.).
i=1 i i

vanishing on V with

L>O if x,EE+UG+,
1. V V

x,<o if x,es'uc’.

l i v v

By a well known result for Haar systems [3 , p. 74], the xi's

must alternate in Sign. Thus o(xi) = (-l)i+lo(x1).

(Sufficiency) Suppose such a v exists. Then con-

sider the matrix M

[w1(x1) w2(x1) . . wn(x1) 0 or ('1)1

w1(x2) w2(x2) . . . w (x2) 0 or (4)2
I . .
M = .

co. n

n+1

 

 

hw1(xn+1) w2(xn+1) . . . wn(xn+1) O or (-1)
where the element in the last column is 0 if x1 E Gv ~ E

and (-1)1 if x1 E Ev’ and at least one element is non-zero.

V

The system

00

()\1,...,)\n ) M '-

+1

HO'°’

has a solution {xi}::l such that not all the A: are zero.

Construct the linear functional

n+1
L(h) = z ih(x.)
i=1 i i

vfliich vanishes on V and for which the Ai's alternate in

n+1

n+1 +1
Sign“ So, by choosing {xi}i=l to be [liligl or i'xilzgl

28
so that

. + +
xi > O for X1 E Ev_U Gv
and

“<0 for x,EE-UG-,
i i v v

Theorem 1.3 implies v is a best restricted approximation

for f. II

If f 6 6(5) we can get some idea of the size of
Hf - VHK for all v E V1 in terms of “f - V*HK and
“V - V*HK for v* a best restricted approximation to f.
E.W. Cheney [3, p. 80 ] proved a theorem relating these
quantities in the standard Chebyshev approximation problem
with V a Haar subSpace, and also for V a set of generalized
rational functions. Similar theorems for the case of restricted
approximation were proved by G.D. Taylor 23] for V a linear
Haar subSpace, and by Leah and Moursund Q6] for V a restricted
set of generalized rational functions. We Shall assume V1

satisfies condition H.

Theorem 1.6: (Strong Uniqueness Theorem) Let V be an n-

~ *
dimensional Haar subSpace, f E C(S). Further let v be the
unique best restricted approximation to f from V1. Then

there exists a constant y > 0 depending only on f such that

for any v E V1,

\If - VII, 2 If - v*uK + yuv - an,

29

‘ggggfg (If Hf - V*HK = O, we can take y = 1 since
IIf - AK 2 W - A, + If - v*IIK = W - VIIK')

*
If v E v , the conclusion of the theorem holds for
*
any positive number y, so we shall assume v i v . Since
*
v is the best restricted approximation to f, there is a

characterizing linear functional L based on points

n+1

{x.} B C E U G ,
i i 1 v* v*
n+1
L(h) = I Bih(xi):
i-l
with
+ +
sgn Bi=cl=+1 for xiEE‘kUGv'k,
sgn Bi = g, = -1 for xi 6 E'* u G-*,
1 v v
n+1 *
and {xi}i=1‘1 E * ¢ ¢. We shall define the function sgn (°)
v
as follows:
sgn y if y i O
*
sgn (Y) =

+1 if y = 0.

Then for xi 6 G * and V 6 V1,
v

* * 'k *
88H (V - V )(xi) = $8“ (f - V )(Xi).

*
Consider the set

x
U = {v E V: °i(v - v )(xi) 2 O for all xi E Gv*}.

Notice that U is closed and we have shown V1 C U. For

an}

sir

11-]

we

And

Ve-

It;

Sinc

Set

30
*
any V€U~[V}:

*
max gi(v -v)(x.) >0
x.€E 1
1 *
v

x *
since L(v - v ) = 0 and v - v cannot have more than

n-l zeros. Thus if

*
ai(v v )(xi) 0 for all xi E G *,

V

we mus t have

*
oi(v v )(xi) < O for some xi E E *.

And if

o.(v

si-
1 v)(xi)>0 for some x,EG

we mus t have

*
v )(xi) < O for some xi E Ev*.

oi(V
It follows that there is a number y > 0 Such that

min max oi(v* - v) (xi) = y > 0
*
Hv -vHK=1 xiEE *
v
VEU

Since it is the minimum of a positive function on a compact

*
set, Now let v E V1 ~ [v ], then

m a: V*(x) - {(1 - +)v*(x) + +— v(x)}.
IIv -VIIK IIv -VIIK IIv --vIIK

Let

31

l

 

 

 

 

 

 

 

 

 

 

vo(x) = (1 - * )v*(x) + * v(x).
HV 'VHK HV 'VHK
Let xi E G-*, that is v*(xi) =p.(xi), then
V
vo(xi) - v*(xi) = - *1 v*(xi) + * v(xi)
“V 'VHK HV fVHK
s - *1 Mxi) + , moi) -- 0.
HV 'VHK HV 'VHK
and oi < 0 so that
gi(v0 - v*)(xi) 2 0.
Similarly if x1 E G:*, v*(xi) = L(xi) and
vo(xi) - v*(xi) = - * v*(xi) + * v(xi)
HV 'VHK HV "VHK
2 - * mi) + *1 L(xi) -- 0.
NV ‘VHK UV 'VHK

and 01 > 0 SO that
* o
x
We have shown that for every v E V1 ... {V I:

IIv-v*IIK

with vo E U and “v0 ' V*IIK = 1' Now for v 6 V1 " {V*I:

= vo(x) - v*(x)

let x, 6 E * be such that
1o v

max C(Xi)(V* - v)(xi) = e<x1L )<v* - v)<xi >.
xiE E * o 0
V

32
Then

Hf - vHK 2 01 (f - V)(Xi )
o o

= 01 (f - "*)("i ) +oi (v* - v)(xi)
0 O O O

2 If - v*uK ...u.* - vuK- II

We define a function T on C(S) = {f E C(S): p > O
and f(x) 2 L(x) for all x E L and f(x) s p.(X) for all
x E J}. T assigns to each f E C(S) its best restricted

approximation v E V Theorem 1.6 easily yields a theorem

1.
which proves T is continuous. This theorem is a logical
corollary to the Strong Uniqueness Theorem and has been proved
by the various authors discussed previously. The original

proof of the continuity of T was done by Borel [l] for the

Standard polynomial case.

Corollary: (Continuity of the Best Approximation Operator)
* ~
Let f E C(S). Then there exists a number I correSponding

* * * ...
to f such that if T(f ) = v E V and if f E C(S) is

1
arbitrary with T(f) = v, then

h-vhxsmf-FL-

Proof: By the Strong Uniqueness Theorem there is a number

y > 0 such that
* * * 'k
\If - VIIK 2 \If - v “K + va - vnK for all v 6 v1.

Thus for any arbitrary f E C(S) and corresponding v = T(f),

33

«Ah - fin, s IIf* - vuK - nf* - v*nK
s If" - in, + If - vIIK - Hf" - v*IIK
s If" - in, + If - v"\\K - IIf* - v*nK
s If" - A, + If" - in, + If" - v*IIK- u£* - v],
= 2n?“ - in...
Now let i, =3; . I

Section 4: Remes Algorithm for Calculating Best Restricted
Approximation

In order to obtain such a best restricted approximation
of a given continuous function f with bounds L and p from
a subSpace V1 of a Haar Space V of dimension n (2 1) we
can modify the algorithm developed by G.D. Taylor and M.J.
Winter [25]. We shall assume that J, K, L are non-empty
subsets of the real line. Thus 8 = JIJ K.U L is contained

in some finite closed interval [a,b]. Assume K contains at

leaSt n+1 points and that V satisfies condition H on

1
[a,b]. Let f E C(S) be the function to be approximated. Then

inf Hf - VHK = p > O.
va1

We shall choose n+1 points of K

and construct a best approximation v1 to f from the full
Haar Space V on these n+1 points. Next, we check to see

if v E V

1 and if Hf - v

1 is greater than If(xi’1) - V1(Xi’1)I

1\\x

34

for i = 1,...,n+1. If Hf - leIK = ‘f(xi’1) - v1(xi,1)I

and v1 E V1, then we are done by Theorem 1.5. If not, we

replace one of the points xi ]_ with a new point from S to
3

get a set

X <X <...<

1,2 2,2 xn+1,2

2. We

repeat this procedure and obtain a sequence of functions

on which we find the best restricted approximation v

{vn} C V (possibly finite) which converge to the best restricted
approximation v E V.

Let

X (X ... X
21< <

1,1 , n+1,1

be distinct points in K, and let

{w1(x),...,wn(x)}

be a basis for V. We further assume that f cannot be inter-
polated on {xi,l}::l by any element of V. This can be done
by selecting any set of n points, interpolating f on these
points and then selecting xn+l,l such that f(xn+1’1) i v(xn+1’1).
This can be done since p > 0. Then the system

n

E ai 1wi(xj 1) +(-l)j an+1 1 = f(xj_1) for j
i=1 3 9 3 9

1,...,n+l

, , n+1 ,
has a unique solution {ai,1}i-l since {wi(x)} is a Haar
system. Set
[1

v1(x) = 121 ai,iwi(x).

35

If “f - vluK = Ian+1,1I = [f(xj) - v1(xj’1)I = el, and

v1(x) 2 L(x) for all x E L,

v1(x) s u(x) for all x E J,'

then by Theorem 1.5, v is the desired best restricted

l

approximation to f from V1. If not, we define the follow-

ing quantities:

M1 = max {v1(x) - u(x): x E J},
m1 = max [L(x) - v1(x): x E L},
E1 = Hf - VIIIK - e1.

Let Y1 = max {E1, M1, m1}. (In case of equality, let Y1 be
the first largest element.) Choose y1 E S in the following
manner:

If y1 = E1, let y1 e x and I£(y1) - v1(y1)I =IIf-v1IIK.

M1.

If Y1 = M1, let y1 E J and v1(y1) ' 9(Y1)

m o

If VI = m1, let y1 E L and L(yl) - v1(y1) 1

We wish to exchange one of the x. s for yl. Define: for

1.1
v E V,
+1, if v(x) = f(x) = L(x) and x E L,
sgn1 (f(x) - v(x)) = -1, if v(x) = f(x) ' u(x) and x E J,
sgn (f(x) - v(x)), otherwise.
We then choose x which is to be replaced as follows:

jo,1
l.a. If y1 S x1,1 and

36
Ssn1(f(x1,1) - V1(x1,1)) = Sgn1(f(y1) - V1(y1));

then replace x1,1 by yl,

i.e. x1,2 = y1
Xi,2 = xi,1 for 1 = 2,...,n+l.
l.b. If y1 s x1,1 and

ssn1(f(x1’1) - v1(x1,1)) * ssn1(f(y1) - v1(y1));

II I!
then replace xn+1,1 by yl,

i.e. x1,2 = y1
xi,2 = xi-l,l for 1 = 2, ,n+l.
2. If xj_1,1 < y1 < xj+1,1 for some j E [l,2,...,n+1]
(Xo,1 = a, xn+2,1 = b) and
ssn1(f(xj,1) - v1(xj,1)) = ssn1(f(y1) - V1(y1));
then replace xj,1 by yl,
i.e. xi,2 = xi,1 for i # j
xj,2 = y1.

3.8. If ylzxn+l,l and

Sgn1(f(xn+1’1) ' V1(xn+1,1)) = Sgn1(f(y1) " V1(Y1));

then replace xn+1,1 by yl,

xi,2

xn+1,2 = y1'

Iie

37

3.b. If y12xn+1,1 and

Sg“1(f(xn+1,1) ' ”1(Xn+1,1)) T sg“1(f(y1) ' V1(y1))‘
then "replace" x1,1 by yl,

i.e. x = x
1,2 i+l,1

xn+l,2 = yl°

Now {x1,2""’xn+1,2} = ({x1,1”"’xn+1,1} " {on,1}) U {YI}

for some jo. We wish to partition the set {x1’2,...,xn+1’2}

J (not necessarily all non-

into three disjoint sets K 2, 2

2' L

empty) in the following way:

x e K if x # y ,

j,2 2 j,2 1
and for jo such that xjo,2 = y1,
let xjo,2 6 K2 if Y1 = E1;
let xjo,2 E J2 if Y1 =‘M1;
let on,2 E L2 if Y1 3 m1.

We continue the process by solving the following System for

n+1
{“J.z}j-1 '
n i
121 o‘j,2wj("i,2) + (’1) an+1,2 ‘ f("1,2) f°r x1,2 6 K2’
['1
iii “J.zwi(xi.2’ = ”(Xi.2) f°r “1.2 6 J2’
n

321 aj,2wj(xi,2) = L(xi,2) for xi 2 E L2.

38

n
Let v2(x) = 1:1 qj’zwj(x), e2 = Ian+l,2‘° If \It - VZHK = e2
and

v2(x) 2 L(x) for all x E L,

v2(x) S.u(x) for all x E J,

then v2(x) is the desired best restricted approximation.

If not, we find y2, y2 in the same manner and continue the

iteration. Suppose we have not obtained the best restricted

approximation but we have found v based on the points

k

x (x <...<X

l,k 2,k n+l,k

and at least one of these is in K. Further we have

Ssn1(f(xi,k) - 36:1,,» = (4)” sgn1<f<x1,k> - vk(x1,k))

for i = 1,...,n+1. If, as before, vk is not the best

restricted approximation to f, then we can find

[11
fl

Hf - v - e

kHK k,

Mk = max {vk(x) - p(x): x E J},

X

mk = ma [L(x) - vk(x): x E L},

Yk a max {Ek’ Mk3 mk} (treat equality as before).

Choose yk E S so that

if Yk Ek. If(yk) - Vk(yk)l = Hf - kaIK;
Yk = Mk, vk(yk) ' “(yk) a Rik;

if Yk = Ink. L(yk) - vk(yk) = Ink.

if

39

Replace xj by yk as described before and form

o,k

{xl,k+l’°°°’xn+l,k+l} = ({xl,k’°°°’xn+l,k] " {xjo,k}) U {Vi}

with

<...< X

x1,k+1 < x2,k+1 n+l,k+l

and partition into three disjoint sets

{xl,k+1’°°"::+l, k+1}

K so that 1U Lk+1le J

k+l’ Lk+1’ Jk+1’ +1 H{ l, k+l""’xn+l, k+l}’

and for x. = x E {x

J,k+l i,k ,k} the“

1,k?”""n+1,kI " {on

xj,k+1 E Kk+1 if xi,k e Kk’

xj,1<+1 E Lk+1 if xi,1< E Lk’
xj,k+1 E Jk+1 1f xi,k E Jk'
F°r j,k+l yk’
Xj,k+1 e Kk+1 if Yk g Ek’
xj,k+1 E Jk+1 if Yk ‘ Mk’
Xj,k+1 6 Lk+1 1f Yk = mk'
Now set v (x) = 210 (x) where [Q }n+1 is
k+l jfl j, k+-1w j j,k+l j=l
the solution of the system
" i
+ _ B
jalaj, k+1Hj(i ,k+1) ( 1) an+l,k+l f(xi,k+1) f°r xi,k+1 E Kk+l’
n
= f
1210’), k+1wxj( i,k+l) ”(xi,k+1) or xi,k+1 6 Jk+1’
n
121?],k+1wj(xi,k+1) = L(xi,k+l) for xi,k+1 E Lk+1°

Kk+l # ¢. Suppose K

40

>) = ('1)i+18gn1(f(x )).

sgn1(f(xi,k+1> ' vk(xi,k+l 1,k+1) ' Vl<(xl.k+1

Thus, if we assume, without loss of generality, that

- = +
Sgn1(f(xl,k+l) vk.("1,1c+1)) 1’
1.8. vk(xl,k+1) S L(x1,k+1)
and
S L(xi,k+l) if i is odd
vk(xi,k+l)

v(x) = vk(x)

for some x, xi,k+l s x s xi+l,k+l’ for each i = 1,...,n.
Thus (v - vk)(x) has n zeros. This is a contradiction
since V is an n-dimensional Haar space.

Again we check to see if vk+1(x) is the desired best
restricted approximation and if it is not we continue.

The proof of the convergence of the algorithm results

from a series of lemmas.

Lemma 1.1: If V is an n—dimensional Haar subSpace of

C[a,b], A > O is given, 5 > 0 and

s = {(x1,...,xn) e R“; a s x <...< xn s b, with

1

‘xi - xi+l‘ 2 6 > O for i = 1,...,n},

then there exists C > 0 such that for any v 6 V with

an

Th

ob

941

41

\v(§£i)| s A for some (i1....,§n) e s, we have ML, ...
max {‘v(x)‘: x 6 [a,b]} s C.
In fact there exists an N such that if
n
v(x) = .2 kiwi“),
1=l
then ‘xi‘ s‘N, i = 1,...,n, where {w1,...,wn} is a fixed
basis for V.
Proof: Suppose there is a sequence of functions
n
vk(x) = iElyikwiu) such that for some i0, ‘xiok‘ 4»o as
k a m. Then we can find a subsequence {vj} c {vk} such

that

lxilj| 2 \xijl for i = 1,...,n and all j,

and

‘xilj‘ a m as j aim.

Ai
Then ‘-—J—1 s l and by taking additional subsequences we can

x111

A.
obtain {v } C {v.} with ‘-$L-“~ l. a finite number.
L J x 1
i1;

By assumption, for each k there is an element

£ = (x,, ... x E S with
j 1], , nj)

‘vj(xij)\ s A.

Since S is compact, by taking another subsequence we can

. Mk
obtain {pk} C {VL} With ‘xik‘ a m, ‘XT_—4 » xi and

1
{xk} a x = (x1,...,xn) E S. Now

n
Iuk<xv>l = ‘1231 xikwiocv)! s A + e

lr—v

42

for v = 1,...,n, for any given 6 > O and k sufficiently

large.

n
= . ‘d
Set Mv ‘ 2 kiwi(xv)‘ Then con81 er

 

i=1
n n )‘ik
“15%) 121 "ikwi(xv) " )‘i,k 121 ulk “1(Xv)'

Now for any a > 0 and k sufficiently large,

n A

 

 

ik
\2 w.(x )I >M - 6
i=1 >‘1 k 1 V V
1
“ )‘ik “
Since 2 w (x ) a Z X W (x ).
i=1 )‘1 k i=1
1
But then
ik
”LR 41231 Milk wi(xv)‘ -. Q as j” m’

n

unless MV 3 0. But 2 kiwi(x) I w(x) E V, and if M = 0,
i=1

for v = 1,...,n, then w(x) has n zeros so xi = 0 for

i = 1,...,n. This is a contradiction since *1 = 1. Thus

‘xik‘ is bounded for each vk, 'I

th
Leuma 1.2: If the iteration does not terminate at the k—

step, then

(i) sgnl(f(xi,k+l) ' vk(xi,k+l)) " sgnl(f(xi,k+l) " vk+l(xi,k+l)) ’

i 1,...,n+1,

(ii) Sgnl(f(xi,k+l) ' vk+l(xi,k+1)) a “DHI'Sg‘r‘iUf‘x )

l,k+1

' vk+1(x1,k+1)), i = 1,...,n+1.

(iii) e 2 e

k+l k’

(iv) e 2 max {f(xi

k+l ,k+l) ‘ L(xi,k+1): xi,k+l E Lk+l}’

(vi)

Sgr

by cc

Withc

43

(V) ek+1 2 max {”(xi,k+1) ' f"("i,k+1)‘ xi,k+1 E Jk+l}’

(vi) v is the best approximation to f on

k+l

with respect to V

{x1,k+1’°°' ’xn+l,k+l} g x1<+1 k+l

where

Vk+1 = {v 6 V: v(x) 2 L(x) for all x E (Xk+1f1 L)

and v(x) s u(x) for all x E (Xk+1{1 J)}

and Hf - ka+1 = XEmax ‘f(x) - v(x)‘.

[(+an

Proof: This proof is as in [25] but is included here for

completeness. (i) is proved by induction. First

_ 1+1
sgn1(f<xi,2> - vlcxmn - (-1) sgn1<f<x1,2) - v1(x1,2>)

by construction. Also LzlJ J2 consists of at most one point.

Without loss of generality, we can assume sgn (f(x ) - v (x ) = +1.
1 1,2 l 1,2

1) Suppose L2 U J2 I ¢. If (i) does not hold then

v1(xi,2) 2 v2(xi,2) if i is even and v1(xi,2) s v2(xi,2)

. . . =_ 1+1
if i is odd, Since sgn1(f(xi,2) V2(Xi,2)) ( 1)

2 1

n times and by the Haar condition v2 5 v1. Then since

L2 U J2 = ¢, 61 = Hf - VIHK' Further, since Y1 = max {E1,M1,m1},

sgn1(f(x1,2) - v2(x1,2)). 80 v would meet v at least

we have v1 E V1 and v1 is the desired best approximation.

This contradicts the assumption that the iteration does not
stop. Thus (1) holds.

2) Suppose L2 0 J2 9‘ ¢- If y1 = m1, xio,2 6 L2.
1 +1
0 . - = - o =
If 10 is odd, sgn1(f(xio’2) ‘VICxio’2)) ( 1) l and

v1(xio’2) s v2(xio,2). If sgn1(f(xi,2) - v2(xi,2)) =

44

-sgn1(r(xi 2) - v1(xi 2)), 1 ¢ 10. then v2(xi 2) 2 v1(xi 2)

5 3 9

if i is odd and v2(xi,2) s v1(xi,2) for i even. Again,

counting zeros, v2 5 v1 is the desired approximation contradict-

ing the hypothesis that the iteration does not stop. Other
cases follow in a similar manner.
If (i) holds for k 2 1 and VR is not the best

restricted approximation, consider cases:

3) If Lk+1lJ Jk+1 = ¢, the argument given in 1) above

works.
4) If Lk+1.U Jk+1 # ¢, then for X1 k+l E Lk+1
vk(xi,k+1) S vk+1(xi,k+l)
d
a" for xi,k+1 6 Jk+l
v (x

k i,k+1) 2 Vk+1(x1,k+i)‘

A13°’ Sgnl(f(xi,k+l) ' vk(xi,k+l)) a +1 1f xi,k+1 G Lk+1

and sgn1(f(xi 6 J

,k+1) ' vk(xi,k+l)) . '1 1f x1,k-+1 k+l°

If sgn1(f(x )) # sgn1(f(x

i,k+l) ' vk(xi,k+l i,k+l) ' vk+l(xi,k+l))

for xi,k+l E Kk+l’ we would again have vk+1 a vk which

contradicts the hypothesis, so (i) holds.

Now (ii) is an immediate consequence of (i).

(iii). Suppose that for some k, ek+1 s ek and,
Without loss of generality, sgn1(f(x1’k+1) - vk(x1,k+1)) = +1.
For xi,k+l E Jk+1 we must have i even and
vk(xi,k+l) 2 Vk+1(xi,k+1);

and for x. we must have 1 odd and

i,k+l E Lk+l’

45

vk(xi,k+l) 5 vk+l(xi,k+l)°

So for xi,k+l E Kk+l

\f(xi,k+1) ' vk(xi,k+l)‘ 2 ‘f(xi,k+l) ‘ vk+1(xi,k+l)‘ = ek-l-l
since ‘f(xi,k+l) - Vk(xi,k+l)\ 2 ek 2 ek+1. Now by (i),
‘ d

vk(xi,k+l) S"'1c+1("i,k+1) f°r 1 ° ‘1
and

vk(xi,k+l) 2 vk+l(xi,k+l) for i even.
Again counting the zeros of vk+1 - vk and invoking the Haar
condition, we see that vk+1 a vk' which is a contradiction.
3° e1<+1 > ek'

(iv) and (v) are proven in essentially the same manner,
so only one will be included here. We shall use induction to
prove (v).

If k = l and J = ¢, the conclusion follows. If

2
k = 1 and J2 ¥ ¢, then {xio,2} = J2 for some 10 and

v1(xio.2) ' “(xio.2) = M1 > Hf ' Vina ' ‘31

> -f(xi ,2) "I'Vl(xi ,2) - 62
o o .

So e2 >H,(xi ’2) - f(xi 2).
o 0

Suppose (v) is true for k 2 1. Then if xi,k+1 is
such that xi,k+l E Jk.” Jk+l’ by (111) and the induction
hypothesis,

- f :
ek+l > ek 2 W {“(Xm) “1,13 “1,1. 6 Jk n Jk+l}

‘ max {U(xi,k+l) ' f(xi,k+l) ’ xi,k+1 E Jkn J1<+1}°

46

Jk and Jk+1 can differ by at most one paint. If Jk+1 C Jk’

then Jk+1 = Jk.n Jk+ and (v) is proven by the above state-

ments. If J

l

k+l ” Jk = {Xio,k+l}’

vk(xio,k+l) ' “(XI-Lem“) ‘ ”k > Hf ' VkH ' 8k

- f(x

> vk(xio,k+l) io,k+l) ' ek+1°

Thus

ek+l > “(xio,k+1) ' f(xio,k+l)'
. . . I =
(Vi) USing Theorem 1.5 With K Xk+1t1 K,
' = L ' B d ' l
L xkfil n , J Xk+1 n J, an 1V1 vk+l’ we conclude
that vk+1 is the best restricted approximation to f on
K' from V'. I

1

For convenience of notation, let 2' denote the sum

over those 1 e {1,...,n+1} for which xi 6 Kk; let 2"
S

k
denote the sum over those i for which x, E Lk; 19C 2
i,k

III

denote the sum over those i for which xi k 6 JR.
9

,nggg_£;§; If the iteration does not terminate at the kEll

step, for each R 2 2 there exist constants xlk’°°°’xn+1,k

satisfying
a ' II III
(1) ek 2: likf(xik)+ z likuxik) + z Mk“("1k)’
(ii) z'hflJ - 1.
n+1
(iii) 2 likwj(xik) = 0 for j = 1,...,n,

i=1
(iv) xik sgn1(f(xik) - vk(xik)) > 0 for i = 1,...,n+1,

47

n+1
(v) 2 ‘xik‘ s Ai< m and A is independent of k,
i=1

(vi) 61, - ek_1 '-' Z'Hik\°Hf(xik) ' Vk-1(xik)\ ‘ ek-i‘

+ zu‘xik‘ °‘L(Xik) ' vk—l(xik)\ + z’llhikHMXik) ' v1c-1(xik)"
(vii) ‘xik‘ 2 1 > 0, A is independent of i and k.

Proof: By Lemma 1.2 (iv), we conclude that vk is a best

restricted approximation to f on K' (with correSponding

L', J') from 'Vi. Then by Theorem 1.3 there exists a linear

functional

n+1
L(h) = 2 B.h(x )
i=1 i ik
with x. E D , and
1k Vk
+

B- > 0 if x,k E E U G+',
i i vk vk
B,<o if x. 6F." Us”,
1 1k vk vk

and such that L vanishes on V. Now let xik = 81/2'181‘.

Then
2")\ik‘ = 1’
xik sgn1(f(xik) - vk(xik)) > 0 for i = 1,...,n,
and
n+1
121 xikyj(xik) = 0 for j = 1,...,n.

Thus (11), (iii), and (iv) are valid.

Since Kk # ¢ for all k and

vk(xik) = ““113 f” xik 6 Lk’

5L

48
vk(xik) = ”“113 f" xik E Jk ’
(iii) implies
I H I”
E xikf(xil<)+ z )‘ik’uxikw' 2 x11<‘J”(xil<)

= E'Mk(f<xik> - vk<xip>

E'hik‘ ”("119 ' vk(xik)‘

U
ek ’3 hik‘

e

k.
To prove (v) we shall show that the sequence
a i a at d Thi f i th a
{(xlk"°°’xn+l,k)}k=l s sep r e . ( s proo s e 8 me
as in [25] except for the construction of the function which
gives the contradiction.) Suppose the sequence is not separated,

then we may extract a subsequence for

G
{0:11 " " ’xn+l,j)}j=1
which there exists a grouping of (xlj’°'°’xn+l j) into a +-l
3

groups (a s n-l),

.3000,x. . . ,ooo, . coo, . ,...,X , ,
(le 113)’ (xil+l,j xizj)’ (Xio+1,j n+1,J)

for all j,

such that

1) there exists 6 > 0 so that for any two points

xij and xkj from distinct groups, we have ‘xij - xkj‘ 2 e

for all j. (If there is only one group, this is vacuously
true.)

2) Setting i0 = O, i = n+1

r-

11

{xES: xi SxSx, j},r=~0,1,...,o,
r

49

. * *
then there eXist xo,...,xO Such that
* *
xi +1 j fl Kr and x1 , a x for r = 0,1,...,o, as j a m.
r ’ r+l’J

Due to the continuity

of f, u, L and the compactness of

K, J, L there exists n > 0 such that for x E J n l. we have

either

MX) - f(X) 2 'fl
f(x) - L(X) 2 H-

or

Let 6 = min {e1,n}/2. Let v0 6 V interpolate

f(x*) j: 6
r

* ,
at xr, r = 0,1,...,o where "+” or "-" is chosen as follows:

* * * *
choose f(xr) + 5 if xr E J!) L and f(xr) - L(xr) < n;
x * * *
choose f(xr) - 6 if xr E J!) L and f(xr) - L(xr) 2 n;
* *
choose f(xr) +-6 if xr E L ~ J;
* *
choose f(x ) - 6 if x E J ~ L.
r r
Otherwise choose "+” or "-“ so that

vo(x:) = f(x:) + (-1)r5.

Since a s n-l, vo exists. We also have

( * * f *
v0 xr) 2 L(xr) or xr E L,
* * f *
vo(xr) s u(xr) or xr E J.
There exists a jo such that j 2 jo implies
a r
v (x) s u(x) for all x E [( U I ) n J],
O r=0 j

50

o
vo(x) 2 L(x) for all x E [( U 1r) n 1g,
r=0 .

U
and ‘f(x) - vo(x)l s 61 for all X E [( U Ir

j) n K].

r=0

But this contradicts the fact that vj(x) is the best

restricted approximation to f on {x1j,...,xn+1 j}, Hence,
3

the sequence is separated. Now define a family of functions

”k E V such that
pk(xik) = sgn xik’ i E {1,...,n+1} ~ {i0},

where 10 is the first integer such that xiok G Kk' Now
by Lemma 1.1, there exists a number C > 0 such that

””ka s c. By (iii)

”I

I = _ I
E'xikukbcik) + Z..iikp.k(xik) I: xikukbcik)

or
2"‘lik‘ +z”’|iik\ s s'liiklc,
n+1
and by (ii) 2 p, \sC+ 1 for all k.
i=1 ik

Now consider
ek ' ek-l = z'xikflxik) + mikuxiQ+ EMMkMXik)
' 3'111klek-1
= Z'xik(f(xik) - vk-l<xik)) + 2"iik(t(xik) - vk_1(xik))
+ zmhkm‘xik) ' vk-l(xik)) ' z'ek-lhik‘
g z'hikU‘flxik) ' vk-1(xik)-‘ ' ek-ll
+ 2"‘xik“L(xik) ' VIC-1(xik)‘

+ zmh‘iku“(x1k) " vk-l(xik)‘ '

51

If (vii) does not hold, then by taking subsequences,

there is an index i0 (1 3 i0 5 n+1) Such that
\xij‘do as j—om.
o
By (v), we may again choose subsequences so that

)‘ijdxi 1,910

1. . a O
10]
and

x.. 41x. for i = 1,...,n+1.
1] i

We notice that xi # xj for i # j, i,j = 1,...,n+l, since

{xlj,...,xn+1 j} is separated. Applying (iii) we conclude
’
n+1
121 xiwj("i) = 0 for J = 1. ..,n.
iii
0

But, by the Haar condition, this implies *1 = O for all i

which contradicts (ii). Thus (vii) must be satisfied. II

We can now show that the algorithm described above is

valid.

Theorem 1.7: If the iteration does not terminate after a

finite number of steps, the sequence {vk):=1 converges
*

uniformly to the best restricted approximation v to f

d *
from 'vl, an e converges to Hf - v “K'

k
Proof: (This proof is essentially the same as in [25].) We

*
shall first show that ekxa e s Hf - v “K = p.

52

1) In Lemma 1.2, we proved vk is the best restricted

approximation to f on {xlk"°°’xn+l,k}' Since this is unique

(Theorem 1.4), we must have

*
since v E V and

*
v (x) 2 L(x) for x E {x1k,...,xn+1,k}r1 L,

*
v (x) s u(x) for x 6 {Xlk’°°"xn+1,k} n J.

ek‘< ek+l for all k implies {ek}«4 e s p.

2) By the above bound on e Lemma 1.2 (iv) and (v)

k,
and Lemma 1.1 imply the existence of a number B > 0 such

that “vkum s B for all k. Then since V is closed and
{vk):=1 is bounded in the norm, there exists a convergent
subsequence of {vk} converging to v E V, and “f - v“co = e.

3) It remains to be shown that v E V By Lemma

1.
1.3 (vi) and (vii)

ek ' ek-l 2 1 max {Bk-1’ Mk-l’ mk-il'

Thus lim sup E = 0, lim Sup M s 0, lim sup m s 0. That is,
k k k k k k

v(x) 2 L(x) for all x E L,
v(x) s u(x) for all x E J.

*
Since the best restricted approximation is unique, v = v ,

e = p " Hf -V\\K-

53

Example: Let s -- [0,1] = K, J = [95,1], L = [o,k], v =

polynomials of degree 5 l. v1(x) E 1, v2(x) E x

 

2
f(x) =x p,(x) = l L(x) = O
. [400
IR!)
\ X
1 100
Let x1,1 = 0, x2,1 = 5, x3,1 = 1

01(1) + (12(0) ' 0’3 = 0

01(1) 4' 012(5) + a3 %

01(1) +'02(1) - a3 = 1

30111121011 a1 3 --8l 0’2 :3 1 a3 a -8!-
1 1
Then V1(X) =-'é-+x, 91:5,
E1={max ‘Xz-x+%‘}-§1-=O
x6[0,l]
1 1
M=max {(-—+x)—1}=-—
l
m1= max {0-(--8-+x)}=%
XE[O,%]
Thus y1=m1 y1 =O=x1’1

54

K2 = {x2,2’ x3,2}
J2 = ¢

L = {x

2 1,2}'

Then solve

ll
0

61(1) +-62<0>

a1(1) +‘az(%) +'a3 = %

a1(1) +‘az(1) - 03 = 1
Solution a1 = 0
=2
0’2 6
=_.1.
C'3 6
5 .1
Then v2 (x) = 6 x , e2 — 6
2 5 1 1
E - max x - -x‘} - +'--
2 6 6 144
xE[0,l]
M2= max {%x-1}=-%
x6[%,1]
5
m2= max {O-gx}=0
X€[0.%]
...5__
V2=Ez y2’12
5
x1,3 0 ’x2,3"12 ’x3,3"1
5 .25—
°’1(1)+°’2(12)+°’3 144
+ - 8
61(1) a2<1> a3 1
... .122. =-}.§_ .15.. l
°’1 1""2 204*“3 204 ’83 2 >6

CHAPTER II
CHEBYSHEV APPROXIMATION WITH RATIONAL

FUNCTIONS HAVING RESTRICTED RANGES: CONSIDERATION
OF EQUALITY IN THE BOUNDS

Section 1: Introduction

In this chapter we wish to consider the problem pre-
sented in Chapter I with rational functions as the approx-
imants. We shall assume S = J U K U L is a closed interval
of the real line, with J, K and L compact subsets of S.
K will be assumed to have a sufficient number of points so
that two approximants equal on K, are also equal on S.

This will be stated more explicitly later.

Let P be the set of functions Spanned by {w1,...,ws}
where w1,...,ws are s linearly independent functions in
C(S), and let Q be the set of functions Spanned by
{v1,...,vt} where v1,...,vt are t linearly independent
functions in C(S). The set R of approximants to be con-
sidered will depend on P and Q.

For the first part of this Chapter, as in Chapter I,
‘we shall assume L and M are given real-valued functions

with L continuous on L and u continuous on J, Such that
L(x) < u(x) for all x E J n L.

111 the latter part of this chapter we shall allow

55

56
L(x) s u(x) for all x E J n L,

where L and H are suitably controlled.
In any case we shall restrict our attention to a sub-

set R1 of R where

= {r E R: r(x) s u(x) for all x 6 J, and

r(x) 2 L(x) for all x E L}.

Assuming R1 ¥ ¢, we wish to find r0 6 R1 such that for a

given f E C(K),

Hf - r OKH = inf Hf - rHK
real

5“1|“
'0
0

If such an ro exists, it will be called a best restricted

rational approximation to f on K.

Section 2: The Existence Prdblem

Let P be the set of polynomials of degree less than

or equal to n and let Q be the set of polynomials of degree

less than or equal to m, and let R be given by

n m
= {r(x) = ( 2 a 1x 1/ 2 b. xj): ai,b. (i = 0,...,n; j = 0,...,m)
i=0 j-O 3 J
m
are real numbers and zbjxj > 0 for all x E S].
i=0

It is well known B, p. 154] that a best unrestricted rational
approximation from R exists for every f E C(S). The follow-
ing simple example due to Loeb [14] shows that best restricted

approximations do not always exist.

57

Example 2.1: Let L(x) = -1 for x E L = {O}, u(x) = 0

 

for x E J = {0}, S = [0,1] = K, f(x) = 1, P = {ax +’b: a,b
are real numbers}, Q = {cx + d: c,d are real numbers}.

Now let

 

rk(x) = x 1 for k = 1,2,...
x +~E

rk E R for each R. Let x 6 (0,1] be fixed, then

lim rk(x) = 1.

k4»
Thus the sequence {rk} converges point wise to the function
r(x) a 1 on (0,1] and the continuous extension of r(x)
to [0,1] is r(x) E 1. But then r G R1. Therefore f
does not have a best restricted approximation from R1.

To eliminate the problem in this example, for this
section we shall assume J = L and J contains no isolated
points. Since J is a compact subset of the real line with
no isolated points, it is a perfect set [18, p. 61].

The proofs of the following existence theorems are
similar to the proofs in the standard rational case found in

[3, pp. 154-155]. We shall assume R1 # ¢ in each case.

Theorem 2.1: Let K, J = L be perfect sets. Let P and

Q be the polynomials of degree less than or equal to n and
m reSpectively. For a given f 6 C(K), a best restricted
rational approximation from R1 to f on K exists.

Q 0
Proof: Let {rk}k=l be a sequence in R1 such that

lim “f ' rkHK = 9

Ram

58

where rk(x) = pk(x)/qk(x) with

n .
_ 1
pk(x) _ .E aikx E P
- 1-0
and
m i
qk(x) = .2 bikx 6Q.
i=0

We lose no generality in aSSuming “qkuc = 1 for each R,

thus “qan s 1. Also, for k sufficiently large

Hf-r K<p'i'e.

RH

This implies

HrkHK < UfHK + p + e

and
HPkHK 5 Hrkux ' quHK‘< “fHK +‘D + e-

m n
Then the sequences {‘aik‘}k-1’ {‘bjk‘}k=l are bounded
sequences for each i = 0,1,...,n, j - 0,1,...,m [19, p. 80].

Thus there exists a subsequence of {rk} for which

aik.d 81 for i = 0,1,...,n

and

bjk—obj for j=0,1,...,m

with b # 0 for some j since “q“ a 1. Then
J m

n .
P (x) ” P(X) ‘ Z 8.x1
k i=0 1
and
m j
Q(X)-*Q(X)=2bx.
k 180 j

59

Now q(x) 2 0 for all x E S. So for x E S such that

q(x) i 0, let

Then

lim rk(x) = r(x)

Ram

for all x E S such that q(x) # O.
S = J U K is a perfect compact subset of the real
line so q has at most m zeros in 8. Since for any

6 > 0 and k sufficiently large

Hrk - fHK < P + 3:
we have

\r(x) - f(x)‘ 3 p
for all x 6 K with q(x) # O, and

L(x) s rk(x) s u(x) for x E J = L

implies

L(X) s r(x) s u(x)

for all x E J = L with q(x) ¥ 0.

So

‘r(x)‘ 5 max (“LHL’ “HHJ’ “fux +'P} E M

for all x E S such that q(x) # 0. Let 2 E S be such that
q(z) = 0. Since p E P, q €,Q are continuous functions and

for all x near 2 but different from 2

6O

\p(X)\ S M \q(X)\.

q(Z)

0 implies p(z) = 0. Since p and q are polynomials
with a common zero, they have a common factor. Thus

J
(x-Z) p (x)
25s). g ° with p() e P, <10 6 Q. and 130(2) 1‘ 0’

i
q‘“) (x-z) qo<x>
q0(z) # 0. Since r is bounded for x near 2, (j-i) 2 0.

 

Repeating this argument for all the zeros of q in S, we

* * *
obtain p 6 P, q E Q, q (x) > 0 for all x 6 S and

*
25:31.22).
q(x) q*(x) for x G S such that q(x) ¢ 0,

*
lim 2351-: 2:151- for z E S such that q(z) = O.
X-oz q<X q (x)

* * * *
Thus we have p (x)/q (x) 6 R1 since p (x)/q (x) 6 R and

for x E J = L

L(X)qk(X) s pk(X) s u(X)qRKX) for each R.

so letting k a m

L(X)q(X) s p(X) s w(X)q(X).

(X)

q(x) s u(x) for q(x) # O,

L(X) S
so by continuity

*
L(x) s 2:151'5 u(x) for all x E J = L.
q (X)

*
**
Also H2;-— fHK s p. Thus p /q is a best restricted rational

q

approximation to f from R1. I

61

If we let P and Q take on a more general form.we
must change R slightly in order to insure the existence of
best rational approximations. E.W. Cheney [3, p. 155 ]
proved the existence of ordinary best rational approximations
from the set R described below.

We shall assume the functions {w1,...,ws} which Span
P and {vi,...,vt} which Span Q are two sets of linearly

independent analytic functions and that R is as follows:

R = {r(x) E C(S): r(x)q(x) = p(x) for some

p(x) e P and q(x) 6Q}.

We again assume K, J, and L are perfect sets and that

J=L.

Theorem 2.2: Let K and J = L be perfect sets and P and
Q be subspaces of analytic functions of C(S) of dimensions
3 and t, respectively. For a given f E C(K), a best restricted

rational approximation to f from R exists.

1

Q
Proof: Let {rk}k-l be a sequence in R such that

1

Hf—r —op a8 k—bm;

ka

where pk E P and qk.E Q are such that
rk(x)qk(x) = pk(x) for all x E S.

lJe can again assume “qkna = l for each k, then quHK s l,

and for k sufficiently large

HrkHK < HfHK + p + e

62
and
HthHmkHwh<HmK+p+a

Thus by the compactness of K there exist functions p E P

and q E Q with

aw
1 1 i i

I
Elm

Pk * P '

.n
ll
Mn

Q e b w
k j=1 j j
for a subsequence of {rk}. Now define a function

r(x) = p(x)/q(x) for x E S, with q(x) # 0. Then

r(X)Q(X) = P(x).
and

lim rk(x) = r(x).

k4»

This implies

‘r(x) - f(x)‘ 5 p for x E K
and

L(x) s r(x) s p(x) for x E J = L,
whenever q(x) # 0. Now let

M -- max { anK + 9. ML. num-

Then

\r(x)\ s M.

‘Now let 2 E S be such that q(z) = 0. S = KlJ J is a

perfect set and q is an analytic function so there is a

63

neighborhood N of 2 such that N c S and for x E N,

using Taylor series, we obtain

2 c.(x - z)j

q(X) =
jzv
and
p(x) = z di<x - 2)i

12H
with cvdu # 0. Since r is bounded by M for x E N ~ {2},
v 2 u and
z die-z)i
lim = r(z)

xaz Z cj(x-z)j

is the continuous extension of r to z, with

 

r(Z)q(Z) = P(Z)-

Now lim rk(x) = r(x) for all x E S, thus

kam

L(x) S r(x) S p(x) for all x E J L

and

“r . fHK S p’
so r is a best restricted rational approximation to f

Section 3: Characterization of Best Restricted Rational
Approximations

In this section we will not be concerned with the
existence of best approximations but rather with characteriza-
tions. Thus we shall assume only that P and Q are s,
respectively t, dimensional subspaces of C(S) where J, K, L

are arbitrary compact subsets of the real numbers and

S = J U K U L. We shall also assume L(x) < p(x) for all

x E J n L. Now let R be as follows:

8 t
R = {r E C(S): r(x) = Z a,W,(x)/ 2 b,v,(x) where
- ._ i 1 ,_ J 3
1—1 J—l
31,...,as, b1,...,bt are arbitrary real numbers
(2
and 2 b.v.(x) > 0 for all x E S}.
j=1 J J

Then
R1 = {r E R: L(x) S r(x) for all x E L

and r(x) s p(x) for all x E J}

as before.

For a given f E C(K) and r E R we again denote

l

the set of critical points as in Chapter I.

E: = {x c- x: f(x) - r(x) = Hf - ruK}.
E; = {x e K: f(x) - r(x) = -Hf - rHK}.
_ + -
Er " Er U Er :
+
Gr = {x E L: r(X) = 1.00}.
G; = {x E J: r(x) = p(x)}:
c =G+U G-
1' r r

Then the following theorem holds for restricted rational
approximation. It is a generalization of Theorem 1.2.
Theorem 2.3: Lat f E C(K) and inf Hf - r” 2—2 p > 0. Then

IGR
rO is not a best restricted rational approximation to f if

 

65
there exists a function ¢ E P + raQ such that

¢(x) > O for all x E E+ U 6+ ,
r0 r0

¢(x) < 0 for all x E Ero U Gr0

The converse holds if there exists an r1 E R1 with

r1(x) > L(x) for all x E L
and

r1(x) < p(x) for all x E J.

Proof: If ¢(x) exists, let

¢(X) = P(X) + roq(X),

and

ML. " “-

p0s) + w(x)

qo(x) ' 5Q(x)

We wish to show that for some 5 > 0, r5 is a better restricted

rational approximation to f than r0. Since qo(x) > O on

 

Consider r6(x) = , where ro(x) = po(x)/qo(x).

S, there is a 50 such that qo(x) - 5q(x) > 0 on S for
5 g 50. Set q(x) = sgn (f(x) - ro(x)).

Consider the following open sets:

Hf-roHK
01 = [x E S: f(x) - ro(x) > —_—2__— and ¢(x) > O},
H“ H
02 = [x E S: f(x) - ro(x) < - -—-§2-K' and ¢(x) < O}.
*' - . .
Er0 S: 01 and Er0 : 02. Let 0 — 01 U 02. By continuity,

we can choose 5 small enough so that f(x) - r6(x) has

1
the same sign as f(x) - ro(x) on O for all 5 3 min {51,50}.

66

Now for x E O and 6 S min {51,60},

\f(x> - r6(x)| q(x)(f(x> - r5(x)>

o(X)(f(X) - ro(X)) +"0(X)(rO(X) - r5(X))

 

5 m(x)
\f(x> - ro<x>\ - q(x) (qo-6Q)(x)

< \f(x) - ro(x)‘ S Hf - roHK'

Since 0 is an open set, K ~ 0 is compact and there is an

61 > 0 such that for all x E K ~ 0,
\f(x) - ro(x)| + 31 s Hf - rOHK.
Then for x E K ~ 0,

‘f(x) - r6(x)‘ S \f(x) - ro(x)‘ +"ro(x) - r6(x)‘

 

- - m(x)
S “f roux 61 + 6 ‘(qo'éq)(X)‘

< Hf - roux

H e
o 1 = - -
for 5 S min {50,51, -§fi—} where n :ég (qo éoq)(x).
That is Hf - réuK.< Hf - roHK'

If x E G:', ¢(x) > O and by the compactness of G:
o 0
there is an open set U on which ¢(x) > 0. Then

'5e¢§3) < o for a < a

ro(x) - r6(x) = (00'5Q)(X) “ o’

and thus r6(x) > ro(x) 2 L(x) for all x E U. L ~ U is again

compact and there exists a number > 0 such that

62

ro(x) 2 L(x) + 32 for all x E L ~ U

67

 

 

 

and
6 mo
r x = - +>r x > L x
5‘ ) (qo-aq><x> 0‘ > ( )
62 H
for 5 S min {50, 2M }.
Similarly for x E G; , ¢(x) < 0. So there is an
0
open set V containing G; on which ¢(x) < 0. Then
0
_ _ 6 ¢(x}
ro(x) r5(x) — (qo-Sq)(X) > O for 5 S 50

and p(x) 2 ro(x) > r6(x) for all x E V. Since J ~ V is

compact there exists a number > 0 such that

63

ro(x) S p(x) - 63 for all x E J ~ V,

 

 

and
= 6 mix)
rém (qo'éqflx) + rem < ”(x)
63 n
for 5 < min {50, 2M }-

Thus by choosing a = min {61,32,33} and

- L11
5 < min {50.61, 2M}, we obtain

r6(x) E R1

and
Hi - ran, < M - roux-

Conversely, suppose there exists r1 E R1 with

r1(x) > L(x) for all x E L,

r1(x) < p(x) for all x E J,

and that r2(x) is a better restricted rational approximation

68

to f than r0, i.e.

Hf - rzux < M - roux.

Let r1(X) = p1(X)/q1(X) and r2(X) = 92(X)/q2(x)o

Let H = min q2(x) > 0 since S is compact. If
xES
92(X) + 6p1(X)
r1(x) # r2(x), let r6 - q2(x) + 6q1(x) . For all 6 > O,

q2(X) + 6q1(x) > 0 for all x E S. Thus r6(x) E R. Let

 

¢(x) = [q2(X) + 6q1(x)](r6(x) ‘ ro(X))

= [92(X) + 6p1(X)] - ro(X)[q2(X) + 6q1(X)] 6 P + rgQ-
Since q2(x) + 5q1(x) > O for all x E S,

sgn ¢(x) = sgn (r6Cx) - ro(x)).

6 q1(X)
(r
(q2+6q1)(X)

 

r6(X) - r2(X) = 1(X) - r2(X))-

 

en or < 31)
T“ f 5 {\\q1\L<Wr1-r2\\,,>}’

‘r6(x) - r2(x)‘ < e for any a > 0.
Now Hf - rZHK < Hf - roHK and

r2(x> - ro<x) = (f(x) - room - (f(x) - r260)
implies

+
r2(x) - ro(x) > O for all x E Ero,

r2(x) - ro(x) < 0 for all x E Ero.

The compactness of Er implies there exists 31 > 0 such
0

69
that
\r2(x) - ro(x)‘ > 61 for all x E Ero.

Choose 5 Such that ‘r2(X) ' r6(x)‘ < 61/2. Then

r6(x) - ro(x) r6(X) - r2(x) +-r2(x) - r0(x)

and

r (X) - r (x) > 0 for all x E E+ ,
6 0 r0
r6(x) - r0(x) < O for all x E Er0

+
Now let x E Gr , then
0

1320:) 2 L(X)q2(X)

and

p1<x) > L(x)q1<x).

r(x)qzcx> +'6L(X)q1(X)

(120!) + 6q1(X)
- L(X)
c12(X) + 6q1(X)

q2(X) + 6q1(X)

 

 

So r5(x) >
= Mic)-
Since ro(x) = L(x)

r6(x) - ro(x) > O for all x E G:;.

Similarly if x E G; , then
0

p2<x> s u<x>q2(x>
and

p1(x> < u<x>q1<x>.

p(xmzm + 6u(X)q1(X) q2(X) + 6q1(x)
So r5(x) < q2(x) +'5q1(x) = p(x) q2(X) +-5q1(x) = ”(X)’

 

 

70
and since ro(x) = p(x),

r (x) - r (x) < 0 for all x E G- .
6 0 r0

Thus ¢(X) = [p2(X) + 6p1(X)] - rO(X)[q2(X) + 6q1(X)] is
the desired function.
If r1(x) = r2(x), let ¢(x) = p2(x) - ro(x)q2(x) and

sgn ¢(x) = sgn (r2(x) - ro(x)) since

¢<x> = q2<x)<r2(x) - ro(X))-

on E: and ¢(x) < O on E .

r
O 0

Since r2(x) > L(x) on L and r2(x) < p(x) on J, it

C

We have shown ¢(x) >

follows that ¢(x) > 0 on G: and ¢(x) < O on G; - l'
o 0

Let f E C(K) and r E R1. Again we can say r is

a best restricted rational approximation to f if
+ + - -
(Er 0 Cr) n (Er U Gr) * 6
as in Chapter I:
1 E+'n G“ # E' n G+'# i lies that to
. r r ¢ or r r ¢ mp
get closer to f we would have to take r E R1-
+ - . . _
2. Er n Er ¢ ¢ implies Hf - rHK — 0.

Since we shall again be primarily interested in those f E C(K)

for which this intersection is empty for all r E R1, 18t

5(K) = {f E C(K): p a inf Hf - rHK > o and
rER1

+ + - -
(Er U Gr) 0 (Er U Gr) — ¢ for all r e R1}.

Condition H for the case of rational functions becomes:

71

Condition H: The subspace R of C(S) will be said to
satisfy condition H if there exists an element r1 E R1 such

that

r(x) < p(x) for all x E J
and

r(x) > L(x) for all x E L.

Remark: If R1 contains two distinct elements r1 and r2
and either P +-r1Q or P +-r2Q is a Haar subSpace, then R
satisfies condition H.

Proof: Suppose P +-r2Q is a Haar SubSpace of dimension d.

Let
r1<x> = p1(x>/q1(x),
r260 = p2(X)/q2(X)o

Then
p1<x> + p2<x>

r05.) = q1(x> + q2(x> E 1"

 

Further, since r1(x), r2(x) 2.5(x) for all x E L and

‘r1(x), r2(x) S p(x) for all x E J,

q1<xn<x) + q2(x>:.<x>
(ll-(X) + (120‘)

 

ro(x) 2 = L(x) for all x E L,

q1(X)u- (X) + q2(X)u(X)
q1(X) + q2(X)

 

ro(x) S = p(x) for all x E J,

saith equality occurring if and only if both r1(x) and r2(x)

iJntersect the bounding curve at that point. Since P + r52 is

72

a Haar subSpace containing p1 - 1‘qu = q1(r1 - r2), to can

intersect 1, and p, in at most d-l points. Then construct
a:

* +
= p + r2q E P + er with ¢(x) = +1 for x E Gro,
¢(X) = -1 for x es; [3, p. 78]. Then
0
~k a: +
(p +Lq)(x)>0 for xEGr (here ro=r2=L),

0

ll
'1
II

*- ~k -
(p + p, q )(x) < O for x E Gr (here r0 2 u).
o

+
Now there exists an Open set U in L containing Gr on

x- * 0
which p +6q >0 and an open set V in J containing

_ * 9:
Gr on Which p + p. q < 0. Also we can find positive real
0

numbers such that

61’62

ro(x) 2 L(x) + 31 for x E L .. U,

ro(x) S p(X) - e for x E J ... V.

2
*
There is a 50 > 0 such that for 5 S 50, (qO - 6q )(x) > 0
on S.
Now assume 6 S 60 and let x E U. Then

* '1:
(p0 + 6p )(X) (L qo - étq )(X)

> = L(x).
(q. - 6q*)(x> (qo - 5q*)(x)

 

 

And x E V implies

* *
(pa + 6p )(x) (use - 6m )(X)
<

 

 

* * = p(x).
(<1o - 5‘1 )(X) (<1o - M )0!)

By continuity we can choose 5 sufficiently small so that

*
(p0 +'6p )(x)

 

>{,(x) on LNU
(qo - 6c: )(X)

73

and

*
(pa + 6p )(X)
(qo - 6q*><x>

 

<p.(x) on J~V. I

Now, as in Theorem 1.3, we shall characterize best

approximations by means of a linear functional.

Theorem 2.4: Suppose R satisfies condition H. Thena

 

9:
necessary and sufficient condition for r E R1 to be a best

restricted rational approximation to f E C(K) is that there

exist k (s dim (P + r*Q) + 1) critical points

x1,...,xk in E*UG*

such that {x1,...,xk} n E * 5‘ ¢; and a linear functional L
1'
defined by

k
L(h) = xih(xi>

i=1
*
such that L vanishes on P + r Q and

+ +
xi>0 for xiEE*UG*,
r r

xi<0 for xi€E*UG*.
r 1'

*

Proof: (Sufficiency) Suppose r satisfies the hypotheses

and r is a better restricted rational approximation to f.

Then

Hf - m, < u: - fin,-

74

For pk) and q(x) such that r(x) = p(x)/q(x), consider

p(x) - r*(x>q(x). Now
* *
sgn (p(x) - r <x>q<x>> = sgn (r(x) - r on)

since q(x) > O for all x E S.

* +

r(xi) - r (Xi) > O for all xi E E * (xi > 0),
r
* -

r(xi) - r (xi) < O for all xi E Er* (Xi < O),
* +

r(xi) - r (Xi) 2 O for all xi E Gr* (xi > O),

x -
r(xi) - r (xi) S 0 for all xi E G * (xi < 0).

By hypothesis, at least one xi E E *, so
r

L<p<x> - r*<x)q<x>> > o.

*
This is a contradiction to L vanishing on P + r Q, so r
is a best restricted rational approximation.
* O
(Necessity) Let r be a best restricted rational
approximation to f with corresponding sets E *, G 9:. Let
r r

* * *
{p1,.q..,pn,r q1,...,r qm} be a basis for P +-r Q. Let F

be defined by

I“ = {(z n+m. = . = ,
1,...,zn+m) E R . zi pi(x), i 1,...,n,

* + +
r (x)qi(x),i=1,...,m for xEE*UG*}
r r

N
ll

) € Rn+mf z = -pi(x), i = 1,...,n;

O z o
’ n+m 1

I-‘
U
O

O

N
ll

* - -
-r (x)qi(x), i = 1,...,m for x E E * U G *}.
r r

75

OE ufif) since otherwise, by the theorem on linear inequal-

njes[3, p. 19] there is a vector (c1,...,cn+m) such that

nhn n
X C.Z.>'0 for all 2 E P. But then $(x) = Z c,p,(x) +
, 1 l . l 1

i=1 i=1

m *

chn+fi:qi(x) is positive on E * U G * and negative on
i=1 r r

E *LJG This contradicts Theorem 2.3. But 0 E co(F)

r r

implies there exist k (S n-I-m-l-l) positive constants

81’“”’Bk and k points 21,...,Zk of P such that

Letting
—-» * ‘k
(+1 1f 21 = (P1(xi):°-°:Pn(xi)sr q1(xi)a°°°ar qm(xi))
61 = -o * *
"1 1f 21 - (“p1(xi)a°°°a‘pn(xi)s’r q1(xi):°°°9'r qm(xi))

and x, = 8.6-: we obtain
1 l i

k
L(h) = lih(xi)

i=1
0 O O I O O O *
which is a continuous linear functional vanishing on P + r(2

with

xi > O for X1 E E * U G *,

Ki < O for X1 E E * U G * .

'3 must be in E *. Suppose not; then
r

r E R1, r(x) = p(x)/q(x), we have

At least one of the Xi

for each

+
r(xi) - L(xi) for xi E Gr*,

76

r<xi> = psi) for x, 6 GI.

Indeed, if there is an r E R1 with

+
r(xi) > L(xi) for some xiE G *,

r
then
*
r(xi) - r (xi) > 0
and
*
P(xi) - r q(xi) > 0-
Thus
( (x) * < )) > o
)‘i P i r q Xi
*
so L(p - r q) > O which is a contradiction. I

*
Remark: If P + r Q is a d-dimensional Haar subspace, then

k = d + 1. (For proof see Remark 1 following Theorem 1.3.)

Although Theorem 1.4 has no direct generalization to
the case of rational approximation, the following theorem,
valid in the standard case [3, p. 164] and in ordinary restricted

rational approximation [15] remains valid in our case.

 

Theorem 2.5: Let f E C(K) and r* E R1 be a best restricted
*
rational approximation to f. If P + r Q is a Haar subspace
*-
then r is unique.
~k

Proof: Suppose P + r Q is a Haar subSpace of dimension d
and that ro(x) = po(x)/qo(x) is also a best restricted rational

* *
approximation to f. Then pO - r qo E P + r Q and

1 *
r - r - q (Po - r 90)-

 

77
Since qo(x) > 0 for all x E S, we have

* ~k
sgn (pO - r qo) = sgn (ro - r ).

*
Now the linear functional characterizing r as a best
approximation must be based on d + 1 points by the Remark
*-
following Theorem 2.4. Further, Hf - r0“K = Hf - r “K’ thus

for xEE*,
r

sgn o;,- r*>(x) = sgn [(f - r*)<x> - (f - ro><x>].
i.e.,

+
20 for xEE*,
* r
(r0 - r )(x) _
S0 for xEE*.

r

+ 1!:
Also x E G * implies r (x)
r

L(x) so

*
(r0 - r )(x) 2 o,

- . . *
and x E G * implies r (x) = p(x) so
r

(rO - r*)(x) s o.

+

* . +
But then L(po-rqo)20 Since )(i>0 for xiEE*UG*

r r
and Ai<0 for xiEE*UG However po-rqOEP-l-rQ,
r r

* 9:
thus L(po - r qo) = O and (po - r qo)(x) must have n + 1

*0

__ * *
zeros. Thus po = r q0 and r0 r . I

In the case of ordinary restricted rational approxima-
tion, an alternation theorem analogous to Theorem 1.5 is valid

[15 j . This is also true here.

 

 

 

78

Theorem 2.6: Let S = [a,b], f E C(K) and R satisfy con-
dition H. Let r E R1.
1. If e(x) = f(x) - r(x) has at least 2 + v

alternations (i.e. there are distinct points

xo<x <...<x in [a,b]

1 v'l-l

with at least one of the following holding at each point

a. ‘e(xi)‘ = HeHK, xi E K,
or
b0 r(xl) = L(Xi): xi 6 L9
or
c. r(xi) = p(xi), xi 6 J,
and for

+1 if e(xi) HeHK or r(xi) =L(xi):

0(Xi) =

-1 if e(xi) -HeHK or r(xi) =p,(xi),

v(xi) = (-l)io(xo) also holds for i = O,l,...,v+l) where
v is the maximum number of zeros of elements in P + rQ, then
r is a best restricted rational approximation to f from R1.
2. If r is a best restricted rational approximation
to f, then e has at least 1 + ‘n alternations where T] is
the dimension of the largest Haar subspace of P + rQ.

Proof: 1. Suppose e(x) has 2+v alternations and r is

not the best restricted rational approximation to f from R1.

Then by Theorem 2.3 there exists ¢(x) E P + rQ such that

 

 

 

 

79

+' +
$(x) > O for all x E Er U Cr
and

¢(x)<0 forall xEErUCr.

But e alternates 2 +v times, thus ¢(x) must have at
least l+v zeros since it is continuous. This is a

contradiction, thus r must be a best restricted rational

approximation to f from R1.
2. Let r be a best restricted rational approxima-

tion to E from R1. Let M be aHaar subSpace of P+rQ

of dimension T} with basis ¢1,...,¢n. Theorem 2.3 implies

there does not exist (6 E M with

>0 for all xEE+UG+
r r

e(X)
<0 forall xEE-UG--
r r

Then by the theorem on linear inequalities [3, p. 19],
5 ( ))~ 6 Eu 6+}
Eco ({(Qp1(x),...,cpnx . x r r
U {(-cpl(x),...,-cp,n(x)): x E Er U Gr}).

So, by the theorem of Caratheodory there exist k + l (S 1] + l)

pOints x0 < x1 <...< xk in Er U GI. and pOSitive numbers

50’ . . . ’Bk such that

k
2 3,3,(p,(x.) = 0 for j = 1,...,n
i=0 i i J i
whe =+1 ‘f ee+uo+ and =-1 if x ee'uc’
re 61 1 xi 1‘ r 6i ' i r r'

Then for K- = B 6.,

80

k
E Aim-1&1) = 0 for j = 1,...,n.

i=0

Since k >1} contradicts the Haar condition, k

a well-known lemma for Haar systems [3, p. 74] the
alternates in Sign

alternate in Sign. This means that e

at least I} + 1 times since the sign of )‘i

by the critical point xi 6 Er U Gr. I

and by

's

>‘1

is determined

It was mentioned in Chapter I that a Strong Uniqueness

Theorem holds for rational approximations both in the standard

theory and in the restricted case.

Likewise it is valid here.

The proof uses the following lemma found in [3, p. 165].

at * ~k *
Lemma 2.1: Let r =p/q ER be such that for P+rQ

*
as a subSpace of C(K) we have dim (P + r Q) = d

If p E P, q E Q satisfy

upuK + 1me = M, + MU.

*
P=rq,

and
q(x) 2 O for all x E K,

*

*
then p—p and q=q on K.

* *
Proof: If r EO, then p .—

dim Q =
*
that q = q' on K.

*
If r #0, then p

* *
p,p 6 PF) rQ. However,

=0 and p=-.:O.

* * **
=rq and p =rq

=s+t-l.

Furthermore

1. Since “un = “(1*HK and q(x)q*(x) 2 0, it follows

implies

dim (P+r*Q) s dim P +dimQ - dim (Pq t*Q).

81

Thus dim (Pn r*Q) S l, and so p is a scalar multiple of

9*- Since HPHK = \\p*\\K and p(x)p*<x> 2 0. p* = p and
q* =q on K. I

We shall assume that P, Q, K and S are such that

if p1,p2 E P, q1,q2 E Q are such that p1 :—=- p2, q1 E q2 on

K, then p1 E p2 and q1 a q2 on S. This will be the case
if P, Q are Spaces of analytic functions of C(S) and K

has an infinite number of points or if P, Q are Haar sub-

Spaces of C(S) and K contains at least maximum {8,t}

points. We shall assume K is a perfect set (this implies

K contains an infinite number of points).

*
Theorem 2.7: Let r be a best restricted rational approxima-

~ 1-
tion to f E 000 from R1. If P + r Q is a Haar subSpace

of C(S) of dimension 3 + t - 1 = d, then there exists a

number y > 0 such that for all r E R1,

uf-nxzh-rhx+nr-3h.

*

Proof: If r E r we can choose any positive number for y.

*
Thus we shall assume r i r . Suppose no such y exists.

Then there is a sequence {rn = pn/qn} S R1 with

ganh-ueJh

“ uJ-rnx

and y -o0 as new.
n

We may assume

hJK+hJK=L

82

Then by the compactness of P and Q, there exist p0 E P,
q0 6Q such that {pm} converges uniformly to p on K,

O

{qn} converges uniformly to qO on K and

HPOHK + “(IOHK = 1-

Setting r0 = po/qo whenever qo 9‘ 0, we have rn —» r0. Also
Yn -+ 0 and
it
2 nrnux - “qu - Hf - r n,

V“ \\r* - run,

 

implies HrnHK and Hr* - rnuK are bounded.
Now r* is a best restricted rational approximation

to f so there is a continuous linear functional L vanishing

*
on P + r Q where

d+l
L(h) = 2 x h(x.)
i=1 i 1
with x1 E E * U G *, at least one xi E E *, and
r r r

+ +
>0 for xiEE*UG*:
r r

h.

i
(Ofor XiEE*UG*o
r r
Let q(xi) = sgn )‘i' Then for r = p/q E R1,
*
q(xi)(p - r q)(xi) 2 0 for xi E G *.

1'

Thus for rn = pn/qn

*
q(xi) (pn - r qn) (xi) 2 0 for X1 E Cr,“

and taking limits

4.
...: l

83

*
q(xi)(Po - r qo)(xi) 2 0 for X1 e Gr*-

Now for X1 E Er*,

ynnr" - run, uf - rnuK - uf - in,

N

o<x1><f - rn)(xi> - q(xi)(f - r*)<xi>

*
= 0(Xi)(r - rn><xi>.

Thus letting n —+ co,

0 2 e(xi><r* - ro><xt)-

Since q0(x) 2 0
at
0 2 0(Xi)(r qo ' Po)(xi)

or

C(Xi)(po - r*qo)(xi) 2 o.

it
But L(pO - r qo) = 0, so

*
q(xi) (pO - r q0)(xi) = O for i = 1,...,d+l

*
and since P + r Q is a Haar subSpace of dimension d, p E r qo.
at

*
p and q0 = q .

Then, using the lemma, we conclude p0
* *
Now qn ... q and q (x) > 0 for all x E K implies there exists

a number 5 > 0 such that qn(x) > 5 for all x E K and n

su fficient 1y large .

If xiEG* and r=p/qER1,wehave

1'

q(xi>(p - r*q>(xi) 2 o

*
and L(p - r q) =0, so

84

*
max o(X.)(r q - P)(X.) >’0
1 i
x,EE
1 *
r
* *

since P + r Q is a Haar subSpace and no (D E P + r Q can
have zeros at all xi E E * U G *. Thus there is a number

r r
c > 0 such that

inf max C(Xi.) (r*q - p) (xi) = c
o=p-r*qu xiEEr*
H¢HK=1

where T is the closed set

T = {p - r*q e P +~r*Q: 0(Xi)(P - r*q>(xi> 2 o

for all xi E Gr*}.

9:
(Notice that if p/q E R1, then p - r q E T.) Now for rn,

let x E E be such that
in r*

'k
xng o<x1><r qn - pn)(xi) = 0(Xin)(r*qn - Ph)(xtn)°
ir*

Then consider the following:

* 1r

ynur - run, Hf - rnuK - Hf - r HR
*

2 °(Xin)(f - rn)(xin) - q(xin)(f - r )(xin)

= C(X.

*
1n)(r - rn)(xin)

* 1
= q(xin)(r qn - pn)(xin) ;;f;:;;

*
2 e(xin)(r qn - pn)(xin)

2 cut-2,, - an,

N

c 6Hr* - rnHK

85

Thatis,yn 2 c 6. But this contradicts Yn a 0. Thus there

is a y > 0 such that

uf-axzh-rhK+Nr-Fh

for all r 6 R1. I

The continuity of the best approximation operator can

now easily be shown as in the polynomial case.

* ~
Corollary: Let f E C(K) with best restricted rational

* *
approximation r E R1. Let P +'r Q be a Haar subsapce of

C(S) of dimension 3 + t - 1. Then there exists a number

8 > 0 such that for any f E C(K) with a corresponding best

restricted rational approximation r,
* *
Hr - rHK 5 SN - fHK-

Proof: For any f E C(K) with correSponding best restricted

rational approximation r, the previous theorem implies

ar-JhsuF-rh-uF-rhp

Thus
nh-rWKsM*-mx+h-rh-uF-rhx
sh*-mK+M-rhK-w*-Jh
s u? - fHK + q: - f*HK .
s.

ur-Jhszwhf-Fh-<e=adx I

86

Section 4: Equalt iy in the Bounding Curves

In all our considerations in Chapter I and in the first
two sections of this chapter, we have assumed that the functions

6 and p. satisfy
L(x) < p(x) for all x E J H L.

In this section we wish to investigate the problem of finding

a best restricted rational approximation if we allow L(x) = p(x)
for some x E J n L. Since results for the case of generalized
rational approximation are valid for generalized polynomial
approximation, we will consider the following problem:

Let S = [a,b] be a finite interval of the real line
and K, J = L perfect sets, Let L(x), p(x) be continuous
real valued functions on J = L with L(x) S p(x) for x E J.

Let P be the subSpace of C(S) Spanned by the s
linearly independent elements w1(x),...,ws(x) and Q the

subspace of C(S) Spanned by the t linearly independent
elements v1(x),...,vt(x). Now for a fixed f E C(K) the
existence Theorems 2.1 and 2.2 are valid for the corresponding

sets R and R S R, when R1 7‘ ¢. No Special properties are

1
required of L and p. other than those imposed in the first
part of this chapter.

However the characterization theorems in the second

section of this chapter required

L(x) < p(x) for all x E J.

87

G.D. Taylor [24] and L.L. Schumaker and G.D. Taylor
[24 considered the problem of existence and characterizations
of best ordinary restricted approximation (where
S = K = J = L = [a,b]) to a given function f E C(S) for the
equality case by extended Chebyshev polynomials, and remarked
that for ordinary rational functions the same results could
be obtained. The concept of an extended Chebyshev system, found

in [8, p. 6 ], is very useful here.

Definition: Let U be the space spanned by n linearly

independent functions p1,...,p.n in C[a,b]. U will be called

an extended Chebyshev system of order p provided “,1 E C(p'1)[a,b],

i = 1,...,n and for all choices ti’ 1 = 1,...,n,
a
StlS StnSb,

(equality can occur in groups of at most p consecutive ti's)

'

1 61(t1) E1(t2) . . - 61(tn)[

fi2(t1) a2(t2) ° ' “a2(tn)
* 1,...,n
U( >=........>0

 

 

t1,...,t
fin<t1> an(t2) . . . Jn(tn) j
where fi,(t,)=p.,(t, if t, <t.;a.(t)=h?3)(t,) if
1 J 1 J J-1 J 1 J 1 J
tj-S-1<tj-s =...= tj,lSiEn.

Let v E U. We say v has a zero of order v (S p-l)

'f = ' =...= (V‘l) = d

at to 6 [a,b] 1 v(to) v (to) v (to) 0 an
v(v) (to) 9‘ 0. We say v has a zero of order at least p at

to E [a ,b] if v(to) = v'(to) =...= v(p-1)(to) = O.

 

88

‘We shall assume P and Q are extended Chebyshev
systems of the same order A, and that L(x) and p(x) are

as follows:

L(xi) = p(xi) for i = 1,...,k;
(let T = {x1,...,xk} c J),

L(x) < p(x) for all x E J ~ T,

and there exists 5 > 0 Such that

m,-l
1 j i-
L(X) a: .2 alj(x ' Xi) " ‘X " xl‘
J=0
for x E [xi - 6, X1 + 5] O J:
m.-l
m.-
i

i .
p(x) = on aij(x - xi)J +~lx - xi‘

for x E [xi - 6, X1 +'6] O J:

k . . . . .
where {mi}i=l is a set of pOSitive integers, {aij : i = 1,...,k;
j = 0,...,mi-l} is a set of real numbers, and let m = 2 m ,
(mi S X for i = 1,...,k).

We wish to consider the set

R = {r(x) = P(X)/Q(X)= P(X) E P, Q(X) E Q, Q(X) > 0

for all x E S},

and

R1 = {r E R: L(x) S r(x) S p(x) for all x E J}.

We shall assume R1 1‘ ¢, r E R1 implies r(j)(xi) = aij for

1:0,ooo,mi-1 and i: 1,...,ko Let

89

01(3) = {f e C(S) ~ R: f(xi) 2 ai , i = 1,...,k}.

O

For f E C1(S) and r E R1, recall

E:'= [x E K: f(x) - r(x) = Hf - rHK},
E; = {x E K: f(x) - r(x) = -Hf - rHK},
G: = {x E J ~ T: r(x) = L(x)},
o; = [x 6 J N T: r(x) = p(x)}.

If for a fixed f E C1(S) and some r E R1,
+' + - -
(Er J Gr) m as1. U Gr) # a.

then r is a best restricted rational approximation since
our previous remarks concerning this case are still valid.

Thus we shall restrict our attention to f E C(S) where

6(8) = {f e c1<s>z (E:U cj) n as; U G? = a
for all r E R1}.

*
For r E R1, consider the set

* .
M * 3 [P + r q E P +-r*Q: (p - r*q)(J)(xi) = 0’
r

i = 1,...,k, j =0,1,...,mi-1}.

. *
(Let the dimension of P +-r Q be d. Then the dimension of

M * is d-m.) The condition (p - r*q)(j)(xi) = 0 is equi-
r .
valent [ 20] to (p/q)(j)(xi) = r*(J)(xi) (the proof proceeds

by induction using p(j)(xi) = [(p/q)q](j)(xi)). M * is a

1‘

subspace of C(S) and if r1 E R1 With r = pl/ql, then

1

 

Illn- .p,'....~ t_ u . I . _ é‘
..Y’. s 1
fl! .
a
, ..._

9O

*
p1 - r q1 E M .k by the form of L and [4,. Each element
r .

m1 at x. for

has a zero of order at least 1

(D E M *
r
i=1,...,k.

The following lemma is a Kolmogorov type theorem. It

will be used to construct a linear functional which characterizes
a best restricted rational approximation.

... ‘k *
Lemma 2.2: Let fEC(S). Let r ER1 with P+rQ an

extended Chebyshev system of order k and dimension d. If

mi < l for i = 1,...,k, and m S d-l, then r is not a

best restricted rational approximation to f if there is an

element (25 E M * with

r

¢(x) > o for all x e a": u 6:,
r r

¢(x) < 0 for all x E E-* U G-*.
r r

* *- ~k
Proof: Suppose such a q) = p + r q exists and let r = p /q .

a-
P
Consider r6 = 2;_+_5_B = J)- . Since q*(x) > 0 for all x E S,
q -6p (‘6

for sufficiently small positive 6 (say 5 S 61),

*
(q - 5q)(x) = q6(x) > 0 for all x E S.

Thus r6 E R. ¢ E M * implies (p + r*q) (j)(xi) = 0 for
r . .
i = 1,...,k, j = 0,...,mi-l; i.e. (p/-q)<J)(xi) = r*(J)(xi)

i=1,...,k and j=

1.1
show that there is a 5 > 0 such that re5 E R1 and r5 is

f than r*.

0,...,mi-l. We wish to

a better restricted rational approximation to

n. P-VWPlll-r [stei- 'CE‘
3 s t . l' I u
.1 .—
DP

 

91
‘k
* \lf-r HK
f(x) - r (x) > —'2——-
*
* Hf-r UK
U[xES: f(x) -r (x)<-——2--

Let 0 = [x E S: and (p(x) > 0}

and ¢(x) < 0}.
Then x E 0 O K implies
[f(x) - r6(x)\ < Hf - r*uK,

andfor xEK~O

[f(x) - r6(x)‘ < Hf - r*nK

for 6 sufficiently Small, say 0 < 6 S 50, as in the proof

*
of Theorem 2.3. So Hf - r5HK < Hf - r HK' Since
*
r6 = £-*—+—§E and P, Q are extended Chebyshev systems of
q ' 5C1
(mi)
is continuous in a neighborhood of Xi’

order l 2 m+l, r6
So, using Taylor series,

m,-l mi
1 j (mi) (X-Xi)
aij (x - xi) + r6 (c) -———(mi)!

i=1,...,k.

r5 (X) = 2
j=0
for some c, xi - 6i S c S xi + 6i and each x E [xi - ei’

+ Ci], i = 1,...,k, we conclude that

x.
i
L(x) S r6 (x) S p(x) for all x E ([xi - ei’ xi + ei] n J)
k
Now setting U = U ([x. - 3., x. + 3,] n S), we can find 5
i=1 i i i i

sufficiently small, say 5 S 52, so that
L(x) S r6(x) S p(x) for all x E (J = L) ~ U.

Then 6 S min [60, 61,62} gives r6 E R1 and

Iw-rwx<w-?h. I

92

~ * *
Theorem 2.8: Let f E C(S), r E R1 and P +-r Q be an

 

extended Chebyshev System of order X and dimension

d = s + t - 1. Suppose also that mi < A for i = 1,...,k

and m S d-l. Then the following statements are equivalent:
1. r* is a best restricted rational approximation

to f.

2. The origin of Euclidean d-m Space belongs to the

convex hull of [q(x)§: 2 E E * U G *} where q(x) = +1 if

+- + r r - -
x E E * U G *, and q(x) = -1 if x E E * U G *, and
r r r r
i = (m1(x),...,md_m(x)) with m1,...,md_m a basis for M *.

r
3. There exists a continuous linear functional

L

m

*
(C[a,b]) based on u d - m +‘l points y1,...,yu

in E * U G *,

r r
u
L(h) = Z lih(yi)
i=1
2 + +
Ki > O for yi E E * U G *,
r r
with
xi < O for yi E E * U G *,
r r
such that L(¢) = O for all ¢ E M *.
r
4. There exist d - m + 1 points
z1 < 22 <...< zd-m+l in Er* U Gr*

such that

i+1

C(Zi)n(zi) = (-1) q(zl)n(zl), i = l,...,d-m+1

m1 mk
where n(zi) = sgn {(zi - x1) (z. - xk) }.

 

93

Proof: (1. = 2.) Suppose 2. is not true. Then by the theorem

 

on linear inequalities [3, p. 19], there is a ¢ E M * with
r
q(x)¢(x) > 0 for all x E E * U G *, i.e.
r r

+ +
¢(x) > O for x E E * U G *
r r

and

¢(x) < 0 for x E E-* U G-*.

r r

*
But then, by Lemma 2.2, r is not a best restricted rational
approximation.

a + . +
(2. = 3.) If 0 E co({(m1(y),--o,¢d_m(Y))3 Y E E * U G *l
r r
U {(-¢1(y),...,-md_m(y)): y E E * U G *}), then by the Theorem
r r
of Caratheodory there exist positive numbers {ai}i=l with

Y S d - m + l and

{aio(yi)w(yi) = o

llM-<

i
Where $01) = (cpl(yi).---.cpd_m(yi))- NOW letting Xi = aio(yi).

we obtain

Y
= ' = ... d-
i: xioj(yi) o for J 1. . m,

1
and

Y
L(h) = z xih<yi)
i=1

is a continuous linear functional on C[a,b] whose null

Space contains M *. If y < d - m.+ 1, then det E¢3(§i)] = 0

r

i,j = 1,...,d-m. where 51 = y1 for i = 1,...,y and

- d-m , . .

[yi}i I1 is a set of pOints in (S a T) ~ {y1,...,yy}.

Then there exist constants Bl"°°’Bd m’ not all zero, with

 

K‘!Wml

 

. 1%....“ {inflafli
, F. ... | ...
..V

 

94

d-m
2 .(§.) = 0.
i=1 Bj¢3 1
d-m d-m
But then T a z Bj¢j E M * has d - m zeros at {yi}i=l
i=1 r

and thus a total of at least d zeros. This is a contradiction

* *
ssince P +'r Q is an extended Chebyshev system and M * C P +'r Q.

r
Thus y = d - m + l.
d-m+1
(3. = 4.) We have 2 X.m.(y.) = O for j = 1,...,s-m,
i=1d milJ 1
and y1 < y2 <...< yd_m+1; {yi}i=l C Er* U Gr*. Let
ai = ikil, i = 1,...,d-m+1. ‘We can use Cramer's rule to solve
d-mfll
1:2 010011.993 (yi) = '0’10(y1)CPj(yl)a J = 1:°°°:d'ma

and obtain

= _ i+l . = _
aio(yi) ( 1) 015(y1) (Ai/Al) for 1 2,...,d m+l,

where

¢1(y1) "' ¢1(yi-l) ¢1(yi+l) "‘ ¢1(yd-m+1)

A. = 0 o
l. o

(Pd-mwl) ‘Pd-m(yi-1) (Pd-m(yi+l) Rd-de-mfl)
for i = l,...,d-m+1.

Ai # 0 for i = 1,...,d-m+1 since Ai = 0 implies the
existence of a non-zero function m E P +-r*Q With d zeros
which cannot happen since P +~r*Q is an extended Chebyshev
system.

Now let 21,...,zd_m_1 be an arbitrary set of d - m - 1

consecutive points in S ~ T. Construct the function w(x)

 

95

I

¢1(X) m1(zl) ... m1(zd_m-1)
W) = E E E
(Pd-m(x) (Pd-111(2).) (Pd-m(zd-m-1)

 

*
m E M * and m i 0 since P +-r Q is an extended Chebyshev
r .
system. m has exactly d-l zeros counting multiplicities. P-

m changes Sign at each 21 and at xi if and only if mi

“I. 31.1.. ...." 5 ...

 

is odd. Let 21 = yi+2, i = 1,...,d-m-1. Then

If Xi ,...,xi 6 (y1’y2)’ then

1 L1
m. +ooom.
i1 ifi
sgn A2 = Sen w(yl) = (-1) 1 sgn w(yz)
mi +...'+mi
= (-l) 1 L1 sgn A1
= (w(y2)/n(y1)) Sgn A1.
Now let 21 = y1, 21 = y1+2 i = 2,...,d-m-l. If
x, ,...,x, E ), then
(p(YZ) = 'A39
(p(Y3) = -A2,
and
+0.0 .
mil mJL2
-Ssn A3 = Sen w(yz) = {-1) sgn w(y3)
+0..
"’31 mjtz +1
= H) sgn A2

= «hop/no,» sgn AZ-

96
Continuing in this manner, we obtain

Sgn Ak+1 = ("(yk+1)/W(Yk)) Sgn Ak, k = ls°°°9d'm:

or
(-1) o(yk) = (w(yk+1)/n(yk))o(yk+1).

and thus
o(yk)fi(yk) = <-1)k+lo<y1>n<y1).

(4. = 1.) Assume l. is not true, that is, ro E R1

and “f - roux < Hf - r*“K. Let ro ==polqO with
upon. + hon, = 1. M, + M, = 1. and p, - r*q, e Mr.-
Let x E E *. Since “f - roHK,< “f - r*“K, and

* *

- B - - f-

ro r f r ( r0),
*

we have Sgn (ro - r )(x) = q(x) for all x E E *. Also

1'

r E R

o 1 implies

* +
(ro - r )(x) 2 0 for all x E G *,
r

* -
(ro - r )(x) S 0 for all x E G *-
r

Thus, if we define

*
+1 if ro(x) r (x)

f(x) 2
* * *
sgn (rO - r )(x) = -1 if ro(x) = r (x) = p(x),
*
sgn (ro - r )(x) otherwise,

* *

then sgn (ro - r )(x) = q(x) for all x E E * U G *, and
r r

by 4. there exist points 21,...,zd_m+1 in E * U G * with
r r

-n“_ nr- [...-F

 

97

”1 sgn" (r0 - r*) “1”“?

(-1)

sgn* (ro - r*)(zi)n(zi)

for i = l,...,d-m+l.

*
If ro(zi) = r (21) for i l,...,d-m+l, then

 

* ” O
130 - r (10 — _
Since M * is an extended Chebyshev system and Lemma 2.1
r
*
yields r0 5 r . This is a contradiction since ro is a
*
better restricted rational approximation to f than r . é
* F
So let zi be such that ro(zi) # r (21), zi+l""’zi+t _
*
such that ro(zi+y) = r (21+Y) for y = 1,...,t and
*
ro(zi+t+l) " r (zi+t+l)° Assume le"°°’xj2 6 (zi’zi+t+l)’

and that mjl +'...+mj2 is odd, 0(zi)‘= +1 and t is odd.

t+l . .
Then C(zi+t+1) = -(-l) q(zi) = -1. 0(21) = +1 implies
+ +* +
ziEE*UG*. If 216E“
r r r

* ‘k
f(zi) - r0(zi) S Hf - roux < Hf - r “K = f(zi) - r (zi).

* + *
Thus ro(zi) - r (Zi) > 0. If 21 E Gr*, r (Zi) - C(21)
*
and ro(zi) > L(zi), so ro(zi) — r (2i) > 0. Similarly

I o *
0(zi+t+l) = -1 implies ro(zi) - r (2i) < 0. Also

(r ' r*)(j)(x ) = 0 for ' = 0 m -1 i = - .
0 i J ’00-, 1 a J1,...,JZ.
*

Thus counting multiplicities, ro - r has a total of at

+. O O Q . o o O . .
least mjl +-sz +~t zeros in (zl,zl+t+1) This is an
*
even number. If ro - r had no other zeros 1n (zi’zi+t+l)

we would have 0(zi) = C(zi+t+l)° Since this is not true we
must have at least mal +...+-mj2 +-t +-1 zeros in

z. z, . e ases s ’ e sa es t at is
( 1, 1+t+1) 0th r c mu t give th me r ult, h ,

98

if ro(zj) ¢ r*(zj), r0(zj+v) = r*(zj+y) for y = 1,...,w

), and if x ,...,x E (z

* 1 ‘2

then r - r has at least m +...+ m +-w + l zeros in
° 1 42

This is also true if w = 0.

*
and ro(zj-+w+l) * r (zj-+w+l j’zj-iw+l)

(zj ’zj+w+1)’

Now let io be the least positive integer such that

*
r(z,)#r(z,) andlet i
0 i0 i0 1
*
such that r (z, ) # r (z, ). Then if x. ,. ,x. E (z. ,z, ) I
0 i 1 J1 32 i i
1 1 o l
and the rest of T is exterior to (zi ,zi ), looking at
o l
subintervals as above if necessary, m, +...+ m, + (i - i )
J1 J2 1 o
* .
zerosare interior to (zi :21 L This means that ro - r 'has 3’

o 1
a total of d zeros, counting multiplicities, in [a,b].

be the greatest positive integer E“

 

* *
Now consider po - r qo E M *. We have shown that pO - r q

r
has a zero of multiplicity m1 at each xi and from the above

0

*
discussion, counting other zeros as simple zeros, (pO - r qo)(y) = O
*
whenever (rO - r )(y) = 0 since qo(y) > O for all y E S.

*

h r has (1 e 08 B t '- P
- Z c r
'1' us p q r u p r q E + Q an

extended Chebyshev system of dimension d which implies

*
p0 - r qO E 0. We have already shown this to be a contradiction. II

Theorems concerning the uniqueness of the best restricted
rational approximation in the equality case described here differ
only slightly from the same theorems in the inequality case.

The simple modifications needed for their proofs will be mentioned
but the details will not be carried out. (For uniqueness results
in a more general setting where the forms of L and u are

not Specified, see L.L. Schumaker and G.D. Taylor 92].)

99

N *
Theorem 2.9: Let f E C(S) and r E R1 be a best restricted

 

*
rational approximation to f. If P +-r Q is an extended
Chebyshev system of dimension d and order k with mi S X

*
for i = 1,...,k and m S d-l, then r is unique.

Proof: This follows in the same manner as the proof of

 

Theorem 2.5. It is necessary only to note that if r0 = po/qO E R1, F7
* * 3
then pO + r qo E M * c P +-r Q, and that the linear functional 1
r 1
L whose existence is given by Theorem 2.8 vanishes on M *. II 1
r
Both in Chapter I and Section 3 of Chapter II, we found ‘5

it necessary to add another restriction, condition H, to the
set of approximants in order to say that the set of points yi,
on which the characterizing functional L depends, included

a point of E *. So far in this section we have not made such
r
an assumption but neither have we any guarantee that one of the

yi's described above is in E *. This will be necessary for

r
the proof of the Strong Uniqueness Theorem given here, so we

introduce condition H'.

Condition H': The set R will be said to satisfy condition H'

 

if there exists an r E R1 with

L(x) < r(x) < p(x) for all x E (J = L) N T.

Lemma 2.3: Given the hypothesés of Theorem 2.9 and R satisfy-
ing condition H', then the set of points in 3. of Theorem 2.8

on which L is based must contain at least one element of E *.
r

then for each r E R1,

Proof: If all the yi's are in G *,
r

r(x) = p(x)/q(x), we have

100

+
r69 = 4,6,) for yi e c ..
r
and

p(yi) for yi e 6-,-
r

r(yi)

Since if for some yi E G-*
r

r(yi) < hop = 36,).

then
( * > o
r yi) - r (yi <
and *
p(yi) - r q(yi) < 0.
SO

ki(p - r*q)(yi) > 0

*
which gives L(p - r q) > 0, but this is a contradiction

*
since p - r q E M * and thus
r

*
L(P - r Q) = 0-
But since H' is satisfied, there must be at least one yi E E *. I.
r
*
Theorem 2.10: Let r be a best restricted rational approxima-

tion to f E C(S) from R and let R satisfy condition H'.

l
*

If P +-r Q is an extended Chebyshev system of dimension

5 + t - 1 = d and order X and mi < X for i = 1,...,k

and m S d-l, then there exists a number y > 0 such that

for all r E R1,

mf-nxzw-rhK+Nr-Fh.

The proof is the same as in the inequality case,

Theorem 2.7, and again we obtain the continuity of the best

 

101

restricted rational approximation Operator.

Corollary: Let f* E C(S) with best restricted rational
approximation r* E R1. Let dimension (P +-r*Q) = s + t -l = d
and P +-r*Q be an extended Chebyshev system of order X,

mi < A for i = l,...,k and m S d-l. Then there exists a
number 8 > 0 such that for any f E C(S) with a corresponding

best restricted rational approximation r,

Hr" - r11, 2 an? - qu.

Comments: 1. In Chapter I we can consider ordinary unrestricted
approximation by choosing L = J = @, or regular restricted
approximation by letting J = K = L. One-sided approximation
can also be considered by choosing either L or u equal to
the function to be approximated and the apprOpriate J or
L = K and the other to be the empty set.

2. In Chapter II the assumption J = L was used to
show the existence of best restricted rational approximations
by bounding the sequence {rn} for which Hf - rnHK a p.
The same reSult is obtained if we assume J C K, L C K or
J ~ K = L N K. In this way we could consider usual un-
restricted rational approximation or one-sided rational approxima-
tion. However if we do not assume J, L are perfect sets we
cannot guarantee existence (see example 2.1).

3. The results of Section 4 of Chapter II can be
obtained with arbitrary compact subsets J, K, L of the real

line if Q = span {1} since existence of best restricted

102

approximations follows from compactness considerations in
this case. We would assume P to be an extended Chebyshev
system of order x (2 mi for i = 1,...,k) and dimension

d (2 m + l). Interpolation and approximation can then be

considered.

CHAPTER III

NON-LINEAR CHEBYSHEV APPROXIMATION
WITH SIDE CONDITIONS

A very general treatment of Chebyshev approximation
with side conditions was given by Karl-Heinz Hoffmann in his
doctoral thesis [7]. In this chapter we shall present an
expository discussion of his work. Some of the reSults of
Chapters I and II can be obtained using the theory presented
here,namely the Kolmogorov theorems and the characterizations
of best approximations by continuous linear functionals.
However, in obtaining results applicable to so many different
problems, some practicality is lost. For example the unique-
ness theorem presented in this chapter is difficult to apply
to any Specific problem and the uniqueness theorems obtained
in Chapters I and II are not results of this work.

Any unreferenced result in this chapter is taken from

the thesis of Karl-Heinz Hoffmann.

Section 1: Definitions and Statement of the Problem and
Standard Theory

We wish to consider approximating a continuous func-
tion f which maps a compact Hausdorff Space Q into a
Hilbert space H. Let C[Q,H] denote the set of continuous
functions from. Q to H with the topology induced by the

uniform norm,

103

 

....I...bev.. 3L. ... Hi

[I
fly}. E t
«..th V. .. lew
h . r.

104
Hf“ = max uf(x)nH,
XEQ
and let E be a Banach Space. We shall assume that there
is an open subset P of E and a continuous function F

such that

F : P a C[Q,H],
and for m E P we shall denote

F(”) = V(°.M) E C[Q.H]-

Now let V = {v(-,m): m E P} be the set of approximating
functions.

We nay further restrict the set of admissible approx-
imants to a subset of V whose elements satisfy given side
conditions. Let K be the scalar field for the Hilbert
space H. We shall assume that K is either the reals or
the complex numbers. Two kinds of side conditions are con-

sidered. Let

f.: P 4 K ' for i = 1,...,k,

(3)

ll
H
v

g : Qj X P a K for j ..,k',

where the sets Qj are compact Hausdorff sets. Define

V1,0 = {v(o.a) e v: fi(fl) = 0. i = 1..--,kl:
_ (J) ,
V0 1 - [V(°,fl) E V: Re gj(t ,fl) 2 0, for all
t(j) E Qj, and j = 1,...,k'],
v1,1 = v1,0 n v0,1
v = v.

105

When we wish to refer to any one of these sets without
specifying which one we shall write V B.
(1’:

The problem to be considered in this chapter is

the following:

(T) For a given function f E C[Q,H], we wish to find

v = 170.91) E v such that
0 0 a B

Hf — v0” S Hf - V“ for all v E V0.8,
that is, vo satisfies
Hf - v0\\ = Edam) = vgf Hf - v1\.
0’58

The concept of extremal signatures will play an
important role in the characterization of the v0 described
above. Since we have not required Q to be a metric Space,
the definitions used here differ slightly from the standard
definitions given by B. Brosowski [2].

Let ,2Y be a non-empty set of ordered pairs (e,M)
where e E C[Q,H] with “g“ S l and M ::Q is closed and
non-empty, and elM (the restriction of e to M) maps M
into the unit Sphere of H. We define an equivalence relation

on ,4; as follows:

(€1,M1), (€2,M2) E Li are equivalent if

and =

The following definitions explain precisely the

concept of signatures used in this work.

106

Definition 3.1:

 

1. Let 2 = (e,M) be an equivalence class of EV.
z is called a signature.
2. If 21 = (el’Ml) and 22 = (€2,M2) are two

signatures, we say 21 C 22 if

MICMZ

and

eUM1 = €2‘M2

for any arbitrary members (€1,M1) E 21 and (€2,M2) E 22.
3. z is called an extremal signature for

v(-,m ) E V with respect to V if for any representative
0 a,B a,

(69M) 6 2
min Re (e(x), v(x,m) - v(x,fl » S 0
o
xEM
for all v(-,M) E Va B.

4. If a signature 2 is extremal for every element

v(-,fl) E'V with respect to V it is called extremal
(1:8 0’38 _
for Va E.
When it is clear that we mean 2 is extremal for
v - m E V with res ect to V we shall 'ust sa 2
( . o) 0.8 p one J y

is extremal for v(°,fl) E V .
0,8
The following examples will help to clarify the above
definitions.
Example 3.1: Let Q be the interval [a,b] of the real

line and V be the polynomials of degree less than or equal

to n. For f E C[a,b], let v0 = v(~,mo) be the best

107

approximation to f in the uniform norm. Then by the

Chebyshev alternation theorem, there exist n + 2 points

<...< x S b

a S X1 < X2 n+2

such that for v = O or 1 (fixed)

f<xi> - vow = <-1>Y“uf - v.11.

y+i

Now let M = [x1,... Then

’Xn+2} and e(xi) = (-l)

(e,M) is an extremal signature for v0 E V, since if
min e(X)(V(X) - V (X)) > 0.
o
XEM
v(x) - vo(x) must change sign at least n + 2 times. But
since v(x) - v0(x) cannot have more than n zeros this is

a contradiction.

The next example, due to B. Brosowski [2], shows that
extremal signatures do not always exist.
Example 3.2: Let V = C[Q,H] and 2 = (€,M) any signature.

Now for (e,M) E z we have a E C[Q,H] = V. Since a i 0,

Re (e(X).e(X)) > 0.

Thus the inequality
min Re (e(x),v(x)) S 0
xEM
is invalid for v(x) = e(x) and therefore no signature can

be extremal.

108

The following inclusion theorem makes use of extremal
signatures.
Theorem 3.1: Let f E C[Q,H] with Z extremal for

v0 = v(.,mo) Q V028. If (€,M) E Z and

f(x) - v(x,m0) = e(x)\\f(x) - v(x,uO)HH for all x c M,

then

min “f(x) - v(x,mo)HH S E(f,Va

>2 Hi - v u-
XEM ’B O

Meinardus and Schwedt [17] proved a similar inclusion
theorem for the approximation of real or complex valued func-
tions and this can easily be generalized to the case of approx-
imation with side conditions.

Let the signature z[f] be defined as follows:

Elf] = (e,M[f]) where
Mm = {x 6 Q: Hf<x>HH = nan,

e e df = [e e 032,sz M s 1, e(X) = f: , x e M[f]].

Using this signature, the Kolmogorov criterion can be stated

as:

Theorem 3.2: Let f e C[Q,H], Va 8 c C[Q,H]. If 2[f - v0]
,
is extremal for v = v(o, m ) E‘V then v is a solution
0 0 Q’s O
of the problem (T) for f.
This theorem gives a sufficient condition for v0
to be an absolute or global minimal solution to the problem

(T), i.e., if vo satisfies the hypotheses of Theorem 3.2,

then

109
l
hf - VOH S Hf — V” for all v E VQ’B.
We shall call v0 3 local minimal solution of the
problem (T) if there is a neighborhood (in the relative

topology on Va ) U of V0 such that

O

:B

Hf - vOH S Hf - v“, for all v E U0.

Throughout this chapter we shall not be concerned
with the existence of a solution to the problem (T) but
rather with the characterization of solutions whenever they

do exist.

Section 2: Structure of V and Properties of the Side
Conditions

We wish to assume the Frechet differentiability of
the functions v, fi’ gj with respect to the parameter E.

Thus the following well known definition is in order [5, p. 92].

Definition 3.2: Let X, Y be normed linear Spaces and 2 an

 

open set in X. A function h mapping Z into Y is said

to be Frechet differentiable at a point m E Z if there exists

 

a bounded linear operator Dh(fl)(~) (called the Frechet

derivative) mapping X into Y such that for all b E X

 

uhnu + b) - he!) - Dh(91)(b)\\Y = 0(\\b\\x)

for Hbe a 0.

Consider the following properties:
(D1) The elements v(-,fl) E V are Frechet differentiable

with reSpect to the parameter E at every point m E P. In

110

this case, for each point
{[91] = {DV(- .906:
of {[M] by d[fl].

b E E}

b e E, Dv(',91)b e C[Q,H].

Let

and denote the Hamel dimension

(D2) The functions fi (i = 1,...,k) are Frechet differ-
entiable at every point m E P and

Dfi(QI)(o): E -O K, i = 1’ 0 ,ko
(D3) The functions gj (j = 1,...,k') are Frechet differ-
entiable at each point U E P and

ng(°3m)(‘): E "' C[Qj,K], for j=

where the topology on C[Qj,K]

uniform norm.

Assuming one or more of
find necessary conditions for a
The regularity conditions given

struct functions in V

0,8.
For V(°,910)EV1 1 we
= (j) .
MJ. {c eoj.
J = {1,...,k'},

c_.
ll

1,...,k',

is that induced by the

these prOperties we wish to
local minimal solution of (T).

below will enable us to con-

define

g.(t(j),91) =

J 0 0%

0 [j E J: Mj # ¢}.

Definition 3.3:

 

1. The side conditions

at
910

if they satisfy (D2) and for every 6 E E

(S) are said to be (Rl)-regular

 

such that

111
Df.(m )b = o, i = 1,...,k,
l. 0

there exists a curve w(s) in P (fl(°): [0,1]

l

P, con-
tinuous), Frechet differentiable at the point 5 = 0 with
Frechet derivative m'(0) and a real number 50 E (O,l]

Such that

fi(fl(s)) = 0, for s e [o,so] and i = 1,...,k,

fl (0) = M0
and there exists a real number k > O with
<u'(0) = 1b.

2. The side conditions (S) are said to be (R2);regular

 

at mo if they satisfy (D3) and there exists a b E E such

that

Dfi(fl0)b = O for i = 1,...,k
and

Re ng(t(j),m0)b > O for all j E JO, t(j) E M .

3. The side conditions are called regular at m

if (D2), (D3), (R1) and (R2) are satisfied.
The following example will illustrate these definitions.

Example 3.3: Suppose E is Euclidean (n+l)-space, Q = [0,1],

 

H = reals, V is the set of polynomials of degree less than
n .
or equal to n, and for m = (80,...,an) E E, v(x,fl) = Z aixl.
i=0
For a fixed f E C[Q,H], we require v(x,m) to interpolate

112

f at x ,...,xk. Then let

1
n i
f.(fl) = 3.x, - f x, for ' = l ... R
J iEO l J ( J). J . . ,
and
n i
Df.(fl)b = Z b.x, , for j = 1,...,k,
J i=0 1 J

for b = (b ,...,b ) E E.
o n

Now suppose ”0 E E is such that
fj(mo) = 0, for j = 1,...,k.

Then if ij(flo)b = 0, let

v(s) m0 +'sb

and (R1) is satisfied with so = 1.

Now suppose we further require v(x,fl) 2 f(x) on

[0,1]. Then let Q1 ==Q = [0,1] and

n .
g1(x9m) = )3 3.X1 - f(x),
i=0 1
n i
Dg1(x,91)b = Z‘. b.X .
. 1
i=0

In this case (R2) cannot be satisfied since

n .
= 1
Dg1(x,flo)b .E bix > 0
1-0
H i
a d f. = b,x, = 0
n J(Qlo)b 1:0 I]

are incompatible. However, if we let Q1 C [0,1] ~ {x1,...,xk}

and k g l§l - 1 then (R2) will be satisfied for some b

[8, p. 30].

113

The following lemma guarantees the existence of

functions in v1 1 "close" to a given function v(-,M) E V1 1.
5 3

Lemma 3.1: Let the side conditions (S) satisfy (D2), (D3) and

(R1) at MO. Then for each b E E such that

Dfi(flo)b = O for 1 = 1,...,k
and

(J') . 0')
Re ng(t ,mo)b > 0 for J E J0, t E Mj’

there exists a curve fl(.) in P, Frechet differentiable

at the point 3 = 0, and a real number 31 E (0,1] such that

fi(m(s)) = 0 for s E [0,31], 1

ll
,..1
V
U
W“
U

Re gj(t(j),m(s)) 2 0 for s E [0,31], j E J, C(j) € Qj’

91(0) = ”0’
and

M'(O) = 1b for some 1 > 0.

Proof: By the (R1) regularity, there exist a curve v(s)

and 50 E (0,1] such that
fi(m(s)) = O for s E [0,50] i = 1,...,k,

91(0) = 21o’

and

M'(O) = Rb for some 1 > 0.

Since w(s) is continuous and w(O) = mo,

91(5) - 91(0) = 0(8)-

114

Also g,(t(j),°) is continuous for each t(J) E Qj’ so
J

(j) _ (j)
gj<t .M(S)) gj<t ,mo + sib)

(r(j) (j) (j) (j),91

O

[3:] 391(5)) ' gj (t 3910)] + [8:101 3910) " gj (t

+ 31b)]

0(3) + 0(3).

30 g (t( (j)

J J),m<s>) = gj<t

,flo + 81b) + 0(3), and

Re gj<t(j).m<s>> = Re gj(t(3>,uo> + s1 Re ng<t(j),mo>b + e(s)
by the Frechet differentiability of gj(t(j),-). Then con-
sider cases:

1. j E J ~ JO. This means Re gj(t(j),fl0) > 0 on
Qj which is a compact set. Thus for some Sufficiently

small 81’

Re gj(t(j),fl(s)) 2 0 for s E [0,31].

2. j E JO. M, is compact, so there is an open set
J

U. 2 M, on which
J J

Re ng(t(j),910)b 2 d > 0,

and for some 52 E (0,81],

Re gj(t(j),fl(s)) 2 0 for C(j) E Uj and s E [0,82].
Now Qj ~ Uj is again compact and f(j) E Qj ~ Uj implies
Re gj(t(j),M(s)) 2 0 for s E [0,8

31’

Where S3 6 (0,82] by the same argument used in part 1. II

115

Using this lemma we obtain the following Kolmogorov
type theorem:
Theorem 3.3: Suppose V satisfies (D1) and the side con-
ditions (S) satisfy (D2), (D3), (R1) and (R2). If

v = V(',910) E V

o is a local minimal solution of (T) for

1,1
f e C[Q,H], then for all h e E such that

Df.(m )b = 0 for i = 1,...,k,
1 o
and

Re Dg,(t(j),m )b 2 O for j E J , t(j) E M,,
J 0 0 J
we have

min Re (f(x) - v (x), Dv(x,fl )b) S 0.
xEM[f-v0] 0 0

Proof: The proof proceeds as in the standard case, i.e.

we assume there is a b1 E E satisfying the hypothesis and

such that
Re (f(x) - vo<x>, DV(X,910)b1) > o

for all x E M[f - v0]. Then we construct a better approx-
imation to f using v0, b1, and lemma 3.1. First, since

(R2) is satisfied there is a bO E E with

Dt.(m )b = o for i = 1,...,k,
1 O O

and

Re Dg,(t(J),m )b > 0 for j E J , t(J) E M..
J o o o J
Then, for a > 0 and sufficiently small,

b = b1 + a b E E,
o

116

Df.(m )b = 0 for i = 1,...,k ,
1 0

(j) . (j) .
Re ng(t .mo)b > 0 for J e Jo, t e Mj,

and Re (f(x) - vo(x), Dv(x,uo)b) > O for all x E M[f - v0]
by the linearity of Dv(x,mo)(-). Let U be an open set

containing M[f - v0] and a > 0 such that
Re (f(x) - v0(x), Dv(x,flo)b) 2 2a > O for all x E U.

By lemma 3.1, there is a curve m in P such that

v(-.m<s>> e V1.1’

91(0) =91 ’

O

and

m'(0) = 1b, with 1 >eo.
By CDl).
”v(x,m(8)) - v(x,flo) - Dv(x,mo)(fl(s) - mo)HE
= 0(1'1916‘7) - MOHE),

and since for any inner product Re (a,b) 2 -(HaH)(HbH), we

have

Re(f(X)-VO(X),V(X,m(8))-V(x,flb)) = Re(f(X)-VO(X),DV(X:flO)(”(S)'flo))
+‘Re(f(X)-VO(X),V(X.fl(8))-V(x,mo)-DV(X.MO)(w(S)-flo))

2 Re(f(x)-vo(x),Dv(x,910)(QI(S)-9JO))-O(H91(S) - QIOHE).

Dv(-,m0)b E C[Q,H] for each h e E, and Dv(x,mo)(.) is a
continuous linear operator from E into H for each x E Q.

Thus

117

an(x,mO)(e)n = sup HDv(x.mo)(c>H
\c E=1
s sup max HDV(X.flO)(C)H = SUP HDV(-,mo)(C)H
HCHE=1 xee uan=1

= HDv(°,mO)(-)H-
This implies
Re<f<x>~vo(x),v<x,m<s))-v<x,mo)) 2 18 Re<f<x>-v0(x>,nv<x,mo>b>
+-Re(f(x)-vo(x).Dv(x.mo)(m(8)-m0-1sb)) - 0(HM(s) - ROME)

2 is Re<f<x>-vo(x),nv(x,mo)b) - Hf-vOH-an<-,mo)(->Ho<s> - o(s>

2 2axs — o(s), for all x E U.

Thus there exists a real number 32 > 0 such that s E [0,32]

implies
Re (f(x) - vo(x), v(x,m(s)) - v(x,mo)) 2 has for x E U.

_. | ‘
Now let h = Hf - VOH - max “f(x) - vo(x)JH. h 2 0
er~U
since Q ~ U is compact and M[f - v0] ; U. For 5 a 0,

Hv(x,M(s)) - v(x,mo)HH s HDv(x,mO)<m(s) - “O’HH + e(s)

s As HDV(~,MO)HHbHE + 0(3)-

So for some 0 < 33 s 32 and s E [0,83]

Hv(x,fl(s)) - v(x,flo)HH s 2A5 HDv(-,mO)H~HbHE.

Choose 30 such that

a , h }.
eiuvvwauzubuEz exuvvo eon ubuE

 

 

0 < s < min {3 ,
o

118

Then for x E U,
“f(x)-v(x,u(so))u§ = Hf(X)-V(x,flo)H: - 2Re<f<x>~v0(x>,v(x,m(so))
- v(x,mo>) + Hv<x.R<so)> - v(x,mo>ufi

s Hf - VOHZ - 2 axsO + 4 125g HDv(-,MO)H:

< Hf - VOHZ.
For x E Q ~ U:
Hf(x)-v(x,91(so))HH s \\f(x)-vo(x)\\H + \\v(x,flO)-v(x,21(8))HH

SHE -vo\\ -h+%<\\f -vH.

0|
Therefore v(-,fl(so)) is a better approximation to f than
v(-:mo). This is a contradiction, so

min (f(x) - vo(x), Dv(x,91)b) s o. I
xEM[f-vo] o

The following lemma will be used to prove a generaliza-
tion of the "zero in the convex hull" prOperty of the set of

extreme points in standard Chebyshev approximation.

Lemma 3.2: Let V1 1 satisfy (D1) and gj (j E J) satisfy
3

(D3). Then the family of functionals
8 = ((f(x) - vo(x). Dv(x.910)(-)) e c[E,l<]: x e o} u

i U {Dg.(t(3).m )(-) e C[E,K]: t(3) e M.}]
J'EJ 3 o J
o
is an equicontinuous family with respect to the norm topology

defined on E.

119

Proof: We shall show that {ng(t(J)

t(i)

.mo)( ) e C[E.K]=
E Mj} is equicontinuous. A similar proof shows
{(f(x) ' V0(X), DV(x,mo)(')) E C[E,K]: x E Q} is equi-
continuous and the conclusion follows since a finite union
of equicontinuous families is equicontinuous.

Let 6 > 0 be given and b0 E E fixed. We wish to

find 6 > 0 such that for any b satisfying Hbo - bv < 6,

'E
we have
(JD) (30) .
Hngo(t ,m0)(b)HH < e for all t E MJO
Since
‘ (jo) | (JO)
Ang (t ’”0)(bo ' b>Hu S (imix Ang (t ’mo)(b° - b>HH
o 0 O
t 6M
jo

= Hng (wow)o - b>H s Hng < emo>H Hbo - bHEe

O

 

if we choose 6 , then for lb - b” < 6, ,
l ‘ o J

6
Jo < Hng (-,wb)\ “E O

o
(,0)
Hng0(t .flo)(bo - b)“H < e
(j )
and 5, is independent of t 0 EM, . I
Jo Jo

Remark: The convex hull of an equicontinuous family of

functions is also equicontinuous.

Theorem 3.4 is the main result of Hoffmann's thesis
[7, Thm 1.10, p. 33] and gives a sufficient condition for

v0 E V1,1 to be a local best approx1matlon when V1,1 18

regular. It will be used later to characterize local best

120

approximation when V1 1 satisfies further restrictions.
3

Theorem 3.4: Let satisfy (D1) and let the Side con-

V1,1
ditions (S) satisfy (D2), (D3), (R1) and (R2). If v0 = v(-,flo)

 

is a local minimal solution from V1 1 for f G C[Q,H], then
3
the following are valid and equivalent:
*
(A) In the dual Space E , the weak * closure of the

convex hull of the set of functionals 3,
3 = {(f(x) - v0(x), Dv(x,mO)-) E C[E,K]: x E M[f - vo]} U
i U {Dg.(t(j),m ) e C[E,K]: t‘j) e M.}].
jEJO J O J

and the linear Space ;£ Spanned by the functionals

{Dfi(mo): i = 1,...,k},

have non-empty intersection.

(B) For all b E E with the property that

Dfi(fio)b = 0 for i = 1,...,k,
and
(J)

Re ng(t ,m0)b 2 0 for j E JO, t

we have
min Re (f(x) - v (x), Dv(x,m )b) s 0.
xEM[f~v ] O O
0
Proof: Theorem 3.3 says that (B) must be satisfied if v0
is a minimal solution.

Assume (A) is not true, that is, 3 fl 1:: ¢. Then

 

Ascoli's Theorem [6, p. 64] implies that co(fl) (closure

9':
a

*
with reSpect to the 0(E ,E) topology) is compact in E

121

since for each b E E

H0(b) {(f(x)-vo(x),Dv(x,mO)b) E K: X E le _ V0]}

and

(J) . (J) -
Hj(b) {ng(t ,uo)b E K. t e Mj}, J 6 J0

are compact sets, and in a finite dimensional space the convex
hull of a compact set is again compact. So the convex hull
of the above sets is compact for each b E E.

;£ is a finite dimensional subSpace of E* and is
o(E*,E) closed. Then by a standard separation theorem [5,
p. 147] for convex sets, there is a 0(E*,E) continuous
functional on E* which Strictly separates fl and ;£.
According to the representation theorem for 0(E*,E) con-
tinuous functionals [11, p. 140], there is an element b E E

such that

(J) - (J)
Re ng(t ,Mo)b > 0 for J E Jo, t E Mj
and
Re (f(x) - vo(x), Dv(x,mo)b) > 0 for x E M[f - v0]

and

Dfi(910)b = 0.

But then, by Theorem 3.3, v0 cannot be a local minimum.
This is a contradiction, thus £10 8 ¢ ¢. This proof also
shows (B) = (A).

(A) implies (B) will be shown indirectly, so we

assume that there exists a bo E E such that

Dfi(910)bO = 0 for i = 1,...,k,

122

' . (j) .
Re ng(t(J),9,Io)bO 2 O for J E JO, t ( M.,

J

and

Re (f(x) - vo(x), Dv(x,flo)b0) > O for all x E M[f — v0].

By proceeding as in the proof of Theorem 3.3, we obtain

b E E such that

Dfi(flo)b = 0 for i = 1,...,k,

(j) . (J)
Re ng(t ,mo)b > 0 for J 6 Jo, t e Mj,

and

Re (f(x) - vo(x), Dv(x,mo)b) > 0 for x E M[f - v0].

But then 8 n 11= @. Since the sets Mj (j E J0), and

M[f - v0] are compact, co :3 n i = <5. I

The usual "zero in the convex hull" theorem is a
corollary to the above theorem Since if there are no side
conditions we can set f1(91) E 0. Then £ = {O} and

0 E c063). More particularly, if v(-,m) iS linear in m,

Dv(-,9.Io)bo = V(°,bo +'mo)- v(-.m0)
and

8 = {(f(x) - VO(X), v(x,b) - v(x,mo)) E C[E,K]: x E M[f - vo]}.

We make the following definitions for convenience of
notation:

{[mo] will be the linear subSpace of C[Q,H] con-
sisting of all elements Dv(-,flo)b with b E E.

£1,0[9103 ={Dv(o,910)b e A910]: Dfi(910)b = o, i = 1,...,k}.

{a [m ] is a linear subSpace.
,0 o

123

20,1910] = {Dv(~.mo)be @101: Re ng(t(j),mo)b z 0;

j E JO, t(j) E Mj}- ib 1[910] is a convex cone.

Let a€1,1[9’Io] =3£a,0[mo] fl =€0,1[9101' if£1,1[mo1 18
a convex cone.
If we do not wish to specify any particular set, we
'11 w 't . , 0 l .
wl r1 e £3,8[m01 (a B E [ a 3)
Assume that the Banach Space E is of finite dimension
n. Let the set V satisfy (D1) and the side conditions (S)
satisfy (D2), (D3), (R1) and (R2). Every element in ;£[flo]
can be written in the form

DV(°,%O)b =

1

"MD

1
where b1,...,bn form a basis for E and a1,...,an are

elements of the scalar field K for E. The following

theorem relates the minimal solution and a linear operator

on C[Q,H].

Theorem 3.5: Let v0 = v(-,mo) be a local minimal solution

 

from V1 1 for f E C[Q,H]. Then for (e,M[f — v0]) in

’

2[f — v0] there exist points

x1,...,xr ( r 2 1) from M[f - v0],

(j) (J) .
t1 ,...,tsj from Mj for each J E JO,

and real numbers

and

such that
’ (j) — . — . '
(1) gj(ti ,mo) — O, 1 - 1,000,8j, J 6 Jo,

(ii) pij > 0, i = 1,...,sj; j E JO,

(iii) x. > O , i = 1,...,r,

dim.£a O[9,10] +-l, if H is real,
3

(iv) r +. Z S. s

jEJo J 2 d' [91 + 1 'f u is com 1e
1m.ia,0 o] , 1 . p x,
and

S

H r (<>(u>> j ((3'))

v 2 l. e x. ,Dv x_, - + 2 Z n ,Dg, t, ,fl -

i=1 1. 1 1 O jEJo 1:]. 1J J 1 O

k f .. * 71‘
+-iil viD i($10)- - O E E .

Proof: If V0 is a local minimal solution from V1 1 for
9
*
f E C[Q,H], then byTTheorem 3.4, the q(E ,E) closure of
c061) and the linear space it have non-empty intersection
*

._ *
in E . Thus 0 of the quotient Space E L£ lies in the

convex hull of the set of elements

{c +£: c E c060}.

* *
The dimension of E is n, so E Li, has dimension n - k.
* *
By the Theorem of Caratheodory [3, p. 17], 0 E E is a
convex linear combination of at most n - k + 1 (or

2n - 2k +'1) elements of the form

c+t.t€£.

 

125

' ' = - k.
It follows that the dlmen51on of ia’0[m01 n
Now every element of £1 can be written as a linear
combination of the elements Dfi(mo)(i = 1,...,k). So there

exist points

x1,...,xr E M[f - v0]

(J) (j) .
t1 ,...,tsj EMj,J 6J0

and real numbers xi’ Hi Yi such that

j,
Ki > O, i = 1,...,r;

>0 '=l,... ' ;
H.. s 1 )Sj, J 6 J0

1]
with
d' +
r +-.2 sj S 1m.£a,0[mo] 1
JEJ
0
(or s 2 difllifi.0[mb] +-1 in the complex case)
and
s
r j (J)
.2_311(S(Xi),DV(Xi,910)°) + .2: .z hijngja, mo)-
1—1 JEJO 1=1
k *
+1E1 Yini(mo). = 0 °

r must be greater than or equal to 1 since, by (R2), there

is a bO E E such that

"
C>

1 O O
and

(J) - (J)
Re ng(t ,mo)bo > 0 for J E J , t E Mj.

Finally, gj(t(J),mo) = 0 since t(j) E M -

 

126

Section 3: A Special Class of Non-Linear Approximation
Problems

In this section we Shall discuss a prOperty of V
which will make the Klomogorov criterion both necessary and

sufficient for a best approximation.

Definition 3.4: Let satisfy (D1). Then V is

V036 OMB

called an equibasis system if for every element

 

 

v = v(-,m ) E V , the Signature 2 is extremal with
O 0 01,6
reSpect to V B if and only if it is extremal for the zero
0’:
element with reSpect to {b B[910]. (9,3 are the same for
3

;£ and V.)

Not every set V is an equibasis system as the

following example shows.

Example 3.4: Let the set V consist of all elements of the

 

form
2
v(x,a) = a - 4a (x - %)2

where a is a real number and let v be defined on [0,1].

The linear Space 11%] consists of all elements

Dv(x,%)b = b - 4b(x - %)2

 

where b is any real number. The signature 2 = (e,M) where

M = {0,1},

6 E 6M = {e E €10,111 ‘e(X)\ S 1: 6(0) =‘4-= 6(1)}

 

‘H

127
is extremal for iiE]; that is
. 2
min e(X)[1 - 4(x - %) ]b g 0

XE{O,1}

for all real numbers b. However it is not extremal for

v(x,i> = i - (x - e>2.

 

r-
since for all b # k,
. 2 2 2
mln e(x)[b-4b(x-¥5) -%+(x-%)]£O. I
XE{0,1}
Many familiar sets are equibasis systems. i,

Example 3.5: Let V be a linear SubSpace of C[Q,H], i.e.

 

the functions v(o,m) are linear in m E P and P is a

Subspace of E. Let the side conditions fi’ i = 1,...,k,

be linear functionals. Then V1 0 is an equibasis system.
3

By definition 3.2, we have

V(°9m + b) ' V(°:mo) = V(°:b) = DV(':mO)(b):
0
and

£10210 + b) ' £10310) = fi<b> = Dfiwoflb), i = 1,...,k.

Now let 2 = (e,M) be extremal for v(x,fl0) E V1 0

= {v(-,fl) E V: fi(fl) = O for i = 1,...,k}. Then
min e(x)(v(x,b1) - v(x,flo)) s 0
xEM

for all q_E P such that v(x,b1) E V1 0. For

DV(':mo)b E 1%,0 = {Dv(':mo)b2 Dfi(flo)b = O: i = 1,‘°°9k}:

we have Dfi(fl5X> 0 for i = 1,...,k and

Dfi(mo>b = fi(m0 + b) - fi(m0) = fi(m0 + b).

.. It‘ll. i: .i IIdu-luui
fink»... e .
n I w,

128

Thus v(x,9,lo + b) E V1, , so

0

min e(X)(Dv(x,flo)b - O) s 0
XEM

for all Dv(x,flo)b E {a,0[mo] and Z 15 extremal for 0
with reSpect to i&,0[mo]' Likewise, if we know

min e<x>(Dv(x,mo>b1 - 0) s 0

XEM

for all Dv(x,flo)b1 E ii 0[910], and if v(x,b) E V then

1,0
fi(b) = 0, i = 1,...,k, and

o = fi(b) - fi(mo> = fi(b - RC) = Dfi(flo)(b - m0).

Thus Dv(x,mo)(b - m0) 6 {a C[mo] and

DV(x,MO)(b - mo) = V(x,b) - V(X.flo)

SO

min e(x)(v(x,b) - v(x,flo)) s O
XEM

and z is extremal for v(x,m ) E V . II
0 1,0

Example 3.6: Let ”1,...,um; v1,...,vn be two sets of
linearly independent real-valued continuous functions defined
on a compact metric space Q. For m = (a1,...,a , b ...,b )

m 1’ n
E Rn+m’ let

vala

ai”i(x)
r(x.m> = 1 1 .

lbgik)

 

MD

1

and set

 

129

n+m n
V = {r(x,m): m E R and Z bivi(x) > 0 for all x E Q}.
i=1
We claim V is an equibasis system. Let Z = (e,M) be

. o
extremal for r0 = r(x,mo) in V where m = (31,...,am ,

bi,...,b:). Then

min e(X)(r(X,9I) - r(x,fllo)) S 0
XEM
for all r E V. We must Show

min e(x)(Dr(X,flo)b) 5 O
XEM

+m
for all Dr(x,llo)b e 1119,10] = {Dr(x,9,[o)b: b 6 Rn }, i.e.,
2 is extremal for O with reSpect to £[mo]. By the extension

theorem of J. Dugundji [2, p. 14], there is an element

(e,M) E 2 such that for all x E Q ~ M,

Ie(x)\ < 1.

Now,
min ((€(X) + r (X) ' r (X))(r(X,m ) - f(x)) s O.
o o o
XEM
So the signature 2[(e(x) + ro(x)) - ro(x)] = z is extremal
for rO with respect to V, and, by Theorem 3.2, rO is a

minimal solution for e + rO E C[Q]. Thus Theorem 3.3 implies

min e(x)Dr(x,flo)b s 0,

 

XEM
1.e.,
1 m n
mg; e(x) n o (igl aiui(x) - ro(x).)-:_1 bivi(x) 5 0.
x Z b,v,(x) 1— 1-
i=1 1 1

This says 2 is extremal for 0 with reSpect to also].

 

\E: g... ..-, .
’v

130

Conversely, if 2 is extremal for 0 6 {[MO], then
n

Since 2 bgv.(x) > 0,
i=1 1 l

m n
min e(x) ( 2 a u (x) - r (x) Z b.V.(X)) S 0:
XEM i=1 1 1 0 i=1 1 1

n
and 1f 2 bivi(x) > 0, then for m = (a1,...,am, b1,...,bm),

i=1 [3%

r(x 2940 E V and 1'

min e(X)(r(X.M) - r(x,flo)) S 0-
xEM

 

So 2 is extremal for v0 with respect to V. II

Example 3.7: Let Q be a compact metric Space and V a

 

subset of C[Q,H] with the following property:
To each pair fl, m0 E P and every real number
t E [0,1], there is a parameter w(t) and a continuous

function

g: Q X [0’1] -’ R:
such that

l. g(x,0) > 0 for all x E Q,

2- (1 - tg)V(-.flo) + tSV(-.m) - V(-.M(t)) = 0(t)

for t a O.

Meinardus and Schwedt [17] called such a set asymptotically
convex. We Shall also assume that our set V satisfies (D1)
and has the following two properties:

3. The function w(t), given above, is Frechet-dif-
ferentiable.

4. m(0) = $0.

131

The following prOperty was proven by B. Brosowski [2].
(F) Dv(-,uo>2u'(0> = g(-,0><v(-.m) - V(°,910))

where fl'(0) is the Frechet derivative of w(t) at the
point t = 0. Thus Dv(~,flo)u'(0) is a function from
[0,1] into C[Q,H].

An asymptotically convex set which satisfies 3. and

4. is an equibasis System. Let 2 be extremal for v(-,Mo) E V

 

with respect to V. Then for all (€,M) E 2,
min Re (e(x),v(x,M) - v(x,” )) s O
o
XEM
for all v(-,m) E V. Again using the extension theorem of

J. Dugundji, we can find e(x) Such that
Ie(x)I < l for all x E Q ~ M.

And, as in the last example, 2 = ZKQCX) +-v0(x)) - vo(x)].
So by Theorem 3.3
min Re (e(x), Dv(x,fl )b) s 0
o
XEM
for all D(-,flb)b E iimo]’ and 2 is extremal for 0 with
reSpect to {[QIO].
Conversely, if 2 is extremal for 0, using (F) we
obtain
min R€(€(X) ,V(X,9.I) " V(X,m )) S O
XEM °
for all v(-,M) E V, that is 2 is extremal for v0 3 v(o,m0)

with reSpect to V. II

 

132

In the previous examples, no Side conditions were
assumed. When side conditions are required, we must consider
more localized properties. In the remainder of this section
we shall assume that the side conditions are regular and that
for a fixed ”0 E P and each h E E satisfying (R1), there
is an associated curve Mb in P Such that

s = + s
mb<> m0 ib( )b,
where lb is a continuous, real-valued, positive function,

differentiable at s = 0 and

1b(0) = 0,

xg(0) > 0.
The set of elements m E P which lie on any Such curve is
denoted by 58'

Definition 3.5: Let vo E V . A set W0 is called a

 

 

0’38
neighborhood+ of v in V if
0 a,B
W0 = {v(.,m) 6 Vans: 91 E do}.
Definition 3.6: A set V is called a local+ equibasis

 

 

CY:

system if for every mo E P:
Whenever the signature 2 is extremal for 0 with
respect to 13>B[flo], there exists a loca1+ neighborhood
’

wo c:V of V0 = v(-,mo) such that 2 is extremal for

CY,

vo W1th reSpect to W0.

 

133

+ .
In general, the neighborhood W0 13 not an open
set with respect to the norm topology, but consists of separate

paths in V with vO as origin.

Example 3.8: Let V be a linear subSpace of C[Q,H] and

 

let the side conditions satisfy (D2), CD3), (R1) and (R2).

Let 2 be extremal for 0 with reSpect to {a 1[fl0], that is
’

min Re (e(x), Dv(x,m )b) s O
o
XEM

for all Dv(.,910)b e £1,1[910]. By (R1) and (R2), there is

a curve
21b(s) = 910 + ib<s>b

satisfying the side conditions and

II
E!

215(0) 0

lb<s>

21;, (0)

N
O

xé(0)b with xé(0) > 0.
By the linearity of V,
min (60‘): V(X,m (3)) " V(X,QI )) S O
XEM b O
for all elements mb(s) of ab. Thus 2 is locally+

extremal for v(-,flb) E V Since the argument above is

1,1'

reversible, the set V is a local+ equibasis system.

1,1
A similar argument Shows that a system of rational
functions or a modified asymptotically convex system with

Side conditions satisfying (R1) and (R2) is a local+ equibasis

system.

as; -m‘ ...—1.».
l

 

 

1|! (Ill-II,
I'll lull-I'll 1"

 

 

 

134

 

 

Definition 3.7: Let Va 8 be a subset of C[Q,H]. An
element vO E V is called a loca1+ minimal solution for
C1”
f e C[Q,H] if there is a neighborhood+ w c v of v
0 01,8 0

such that V0 is a minimal solution for f front W0.
We can characterize a best approximation to a func-

tion f E C[Q,H] from an equibasis system.

Theorem 3.6: Let be a Subset of C[Q,H] and an

V1,1

 

equibasis system. The element v(-,mo) E V1 1 is a local

minimal solution for f with respect to V1 1 if and only if
,

min Re (f(x) - v (x), Dv(x,fl )b) s O
xEM[f-vo] O O

for all Dv(°,mo)b E ii,l[mo]'
Proof: This theorem is an immediate consequence of Theorems

3.2, 3.3 and the fact that V is an equibasis System.

1,1

Theorem 3.7: Let V1 1 be a subset of C[Q,H] and a loca1+
,

equibasis system. The element v(-,mb) E V1 1 is a local+
3

minimal solution (with reSpect to a neighborhood+ of v0) for
the function f if and only if
min Re (f(x) - v0(x), Dv(x,fl )b) s 0
xEM[f-vo] O

for all Dv(o,910)b E £1,1[QIO].

Theorem 3.8: Let be a (local+) equibasis system.

V1,1
Then v(.,mo) E V1 1 is a local (loca1+) minimal solution
,

 

* 7':
for f E C[Q,H] if and only if the q(E ,E) closure (in E )

of the convex hull of the set 3,

135

:5 = {(f(x) - vO(X),DV(X,910)°) e GEE,KJ= X € MU ' V01}
0 [jéJ {ng(t(j),mo)- e C[E,K]: J 6 JO. t(j) E Mjlj.
'0

and the linear Space 11 Spanned by the functionals
Df,(m ), i = 1,...,k,
l 0

have non~empty intersection.
.nggfz The necessity follows from Theorem 3.4.

(Sufficiency) We assume that V0 is not a local
(local+) minimal solution for f. Then the signature 2
cannot be extremal for 0 with reSpect to ii,1[fl0]. So there
is an element Dv(-,flo)b E {a’1[flo] such that for all

x E M[f - v0],
Re (f(x) - vo(x), Dv(x,mo)b) > O.

*
But then the intersection of the 0(E ,E) Closure of the
convex hull of the set 3 and the linear space {i must be

empty. This is a contradiction. I

As a corollary to this theorem we have the following

result of Cheney and Loeb [4].

Corollary: Let V be the set of generalized rational func-
tions as in example 3.6 and C[Q] the linear Space of real-
valued functions defined on a compact metric Space Q. Then
a necessary and sufficient condition for rO E V to be a
minimal solution for f E C[Q] is that the zero of Euclidean

(n+m)-Space lie in the convex hull of the set

136

{(f(x) - ro(X))[u1(X)s--o,um(X)s —v1<x>ro<x>,...,-vn<x>ro(x>3:

x E M[f - ro]}.

Section 4: Results Concerning Uniqueness

The following theorem characterizes those equibasis

systems which yield unique approximations.

Theorem 3.9: Let c: C[Q,H] be an equibasis system.

 

V
1,1
Thenthe following assertions are equivalent:
(A) Every f E C[Q,H] has at most one minimal
solutlon 1n V1,1.
(B) For all v = v(°,fl ) E V and every signature
0 0 1,1
2 extremal for v0, the difference v - V0 with v(-,fl) E Vl
’
and v - yo i O is non-zero in at least one point x E M

of the signature 2.

(C) For each f E C[Q,H] ~ V1 and for every best

,1

approximation v0 E V for f

1,1

\Iv - VOII < ZIIf - vOII for all v E v1,1°

(D) For every pair of functions f1,f2 E C[Q,H] ~ V1,1
and for every pair of best approximations v1 = v(-,m1) E V1 1

for f and v = v(-,m2) for f

l 2 2’

IIV2 ' v1II<IIf1 " V2II + IIfz " V1II°

1
with

Proof: (A) = (B). Suppose there exists vO E V1 and an
‘ ’

extremal signature 2 and an element v E V1 1
9

v - vO i O

l

 

137

and

v(x) - vo(x) = 0 for all x E M.

Then, we claim, there is a function f E C[Q,H] ~ V1 1 for
3
which v0 and v are both minimal solutions with reSpect

to V Let

1,1'
m(x) = “v(x) - VO(X)HH for all x E Q,
m(x) E C[Q,H],

E'= max m(x).

XEQ

Let (e,M) E 2- Then define
Mfl=eQHE'Mfl%

s E, for x E Q ~ M,

HMMM
= E, for x E M.

Set f(x) = h(x) +-vo(x). Then V0 is a minimal solution

for f with reSpect to V1 1 with
IIf - voII = K'

Since 2<: Z[f - V0], 2 is extremal for v0 with reSpect to

V and therefore so is z[f - v0]. Then by the Theorem 3.2,

1,1

V0 is a minimal solution. But, v E V1 1 is also a minimal
3

solution Since for all x E Q,

“f(x) - V(x)HH = Hh(x) +-v0(x) - v(x)HH

e NW)“, +1\vo<x> - won,

138

s IIe<x>IIHdE - m(x)) + m(x)

SK.

(B) = (A). Suppose there is an element f E C[Q,H]

with two best approximations vo,v E V1 Then the signature

.1'
A = zif — v0] n zif - v]

is extremal for v0 and v with respect to [2, p. 105].

V1,1

For every point x E MA,

f(x) - v0(x) = f(x) - v(x).
v(x) - vo(x) = O for all x E MA.
But this contradicts (B) unless v(x) E vo(x).

(A) a (C). Let vO E V1 1 be the unique minimal
3
solution for f E C[Q,H] ~ V1 1; that is
3
Hf - VOH < Hf - v“ for all v E V1,1'

Then

IIvo - vu - IIf - vu e If - VII,

IIvO * VII < 2IIf - VII-

(C) = (B). If (B) is not true, we wish to Show there

is a function f E C[Q,H] ~ V with a best approximation

1,1

v E V such that for some v E V

0 1,1 1,1’

IIVO - VII = ZIIf - VII-

139

Note: Hvo - VH - Hf - V“ s “f - VOH g Hf - V“ for all
v E V1,1. Thus Hvo - v“ i 2Hf - VH.
Now let v0,v give a contradiction to (B) with the

signature 2. Let m, K, e be as in the proof of (A) a (B).

Set

h(X) = % e(X)(E - m(X)) + % e(X)(V(X) - VO(X))-

For x E M, Hh(x)IIH = %' since v(x) - vo(x) = O. For

x 6 Q .. M, IIh(x)IIH s 150? - m(x)) + %IIv(x) - v0(x)IIH s %.
Now define f(x) = h(x) +-vo(x). Then the following are
true:

1. Hvo - V“ = ZHf - v“ since “f(x) - V<X>HH

_ =i‘<'/2 for XEM,
= %(K - m(X))IIe(X)IIH

5;K/2 for x E Q ~ M.
2. v0 is a minimal solution for f with reSpect

to V1,1 since

IIf - VOII = IIhII =§

and 2<: z[f - v0], so, by Theorem 3.2, v0 is a best approx-
imation for f.

(C) = (D). Let v be a best approximation to f

1 l

and v2 a best approximation to £2. Then,by (C),

“V2 ' V1” < szl ' V2”,

IIV1 ‘ vzII < 2IIfz " V1II°

Thus

140

IIV1 ‘ V2II < IIfl ‘ v2II + IIfz " VIII-

(D) = (B). Again suppose that there exist a v1
and v0 and a signature 2 extremal for v0 contradicting

(B). Define

how) 32 good? - m(x)) + £2<v1<x> - vo<x>>,
h1<x) = 15 good? - m(x)) + woo) - q(x)).

Then setting

fo(x) ho(x) +-vo(x)

and

f1(x) h1(x) +-v1(x)

will give the desired contradiction to (D).

1. Hvo - v1“ = IIf1 - vOH + “£0 - V1“, since

E72 for x E M,

Hf1(x) - vo(x)I\H = “fo(x) - v1(x)IIH ‘—
s K/Z for x E Q ~ M.

2. v0 is a minimal solution for f .
o

E/Z for x E M,
“f0(x) - vo(x)IIH

s K/Z for x E Q ~ M,
and Z<: z[f - v0]. Since 2 was extremal for v0, v is
a best approximation to £0.
3. Hf1(x) - v1(x)I\H = K/Z as above. So 2': 2[fl - v1].

It remains to be shown that 2 is extremal for v1. We know

141

Re (e(x), vo(x) - v1(x)) = O for all x E M
and

ufin.Re (e(x), v(x) - vo(x)) s O for all v E V

XEM 1,1

But these imply
min Re (e(x). V(X) - VO(X)) S 0
xEM

for all v E V Therefore 2 is extremal for v1. II

1,1’

Remark: Suppose V1,1 is a finite dimensional linear sub-

space of C[Q,H]. The existence of a best approximation for

any f E C[Q,H] is well known. Theorem 3.9 says that every

f E C[Q,H] has exactly one best approximation v0 = v(',fl0) E V1,1

\

if and only if
IIVOII < ZIIfII-
By (C), each f i 0 has a unique best approximation v0 if
IIVO ' VII < 2IIf - VII

Since V so

for all v E V1 1,1

is linear, v E O E V

.1' 1,1’

IIVOII < ZIIfII-

Conversely, if V0 is a best approximation for f, it follows
that vO - v is a best approximation for f - v for each

v E V1 1. Thus

IIVo - VII < 2IIf - VII-

By Theorem 3.9, v0 must be unique.

 

[([J'l‘llll'llll‘l'l'llll'l

142

Section 5: A Special Case

We wish to consider approximation of functions

f E C[Q,H] by elements of a subset of C[Q,H] when

V1,1

the elements of V are determined by a finite dimensional

1,1

parameter space E.

Theorem 3.10: Let the subset of C[Q,H] be an equi-

V1,1

basis system and let the parameter Space E of the elements

of have finite dimension n. Then v0 = v(-,fl0) is

V1,1

a minimal solution from V1 1 for f E C[Q,H] if and only
,

if there exist points

x1,...,xr from M[f - v0] (r 2 l),

tiJ),...,t:%) from Mj, j 6 Jo,

and real numbers {xi}l’ {uij}, {vi}: such that

(j)

i

gj(t ,mo) 0 for i = 1,...,sj; j E JO,

uij > O for 1 = 1,...,Sj; J E JO,

*1 > o for i = 1,...,r,

with

. {d1m.£a,0[mo] + 1 if H is real,
r + Z sj s
o 2 1m_1a,0[flo] + 1 if H IS complex,

and

r

E )\.(€(X.)9DV(X.:QI )°) + Z

,_ 1 1 1 o , ,
1—1 JEJO 1

ll MH-

0

*
+ xini(mo)' - O .

"MW

1

i

 

143

Proof: The necessity has been established in Theorem 3.5.

(Sufficiency) Assume
s

j
xi(e(xi). DV(xi.flO)b) + .2 .2 uingj(t
l JEJO 1=1

(j)

i

,m )b
0

d can

1

+

i yini(i’lO)b = 0

1

"MW

for all b Such that Dv(',fl0)b E ii 1EMO]. By the defini-

tion of ii 1[m0] we conclude

I'
Z x.(e(x.). DV(X.,fl )b) S 0,
. 1 1 1 O
1=1

i.e. min Re (e(x), Dv(x,mo)b) s 0
XEM

for all Dv(x,m0)b E ii,1[flo]. This means £[f - v0] is
extremal for O with respect to ;£1’1[m0]. Since V1,1 is
an equibasis system, z[f - v0] is extremal for v(.,mo)

with respect to V Thus v(o,mo) is a minimal solution

1,1‘
for f e C[Q,H]. ||

An analogous theorem holds in the case that V1 1
3
I + I I O I
18 a local equ1ba51s system which characterizes a local

minimal solution.

Section 6: Relation of Chapter III to Chapters I and 11.

Some of the results of Chapters I and II can be
obtained from the theory of Chapter III. Consider the prob-
lem presented in Chapter I. Here Q = K and H = real
numbers. The Banach space E E P is Euclidean n-space Rn,

and the function F : P a C[Q,H] is defined by

 

 

144

F(fl)

II
"M :3

aiwi(x)

i l

where m = (a1,...,an) e P = R“ and w1,...,wn are linearly

independent elements of C[Q,H] = C(K).
n
V = {v(x,fl) = F(M)= fl 6 R },

V1 1 = {v(x,fl): v(x,fl) 2 L(x) for all x E L

and v(x,m) s ”(x) for all x E J}.

V1 1 is the Subset of V obtained by the restrictions:
,
f1: Rn a R1,
f1(9.1)E 0;
l

g1: LXRn-oR,

glow) = v(x,m> - L(X);

n l

82(Xsfl) = p(X) - v(X.W)~

(The restriction f1(M) E O is stated for convenience and
does not limit the set of approximants, i.e., V1,0 = V.) The
functions v, f1, g1, g2 are all Frechet-differentiable with
respect to the parameter M with the following Frechet
derivatives:

Let b = (bl""’bn) E Rn be arbitrary,

DV(X,M)(b) =

its :

biwioo,

1 1

Df1(91)(b) E 0.

 

 

 

145
n
Dg1(x,m) (b) = 1:1 biwi(X).

n
Dg2(X.91)b = -121 biwi(x)°

Remark 1: The side conditions 8 = {fl,g1,g2} are (R1)-

regular for all m E Rn since

 

Df1(9,[)(b) a o for all b e R“.
Set 91(t) = m + tb, for t 6 [0,1]. Then
E.
f1(9.l(t)) = 0 for all t 6 [0,1] t
and
fl(0) = M
and
91'(0) = (1)b.

Remark 2: The side conditions S = {f1,g1,g2} are (R2)-
regular for all m E Rn if and only if V satisfies con-
dition H, i.e. if and only if there is an element v(x,ml) E V1,1
such that

v(x,m1) > L(x) for all x E L

and

v(x,fl1) < p(x) for all x E J.

Proof: Suppose the side conditions are R2 regular for all
n . n .
M E R . Fix MOE R and let v0 = v(x,fl0), then there IS a

b E Rn such that

Dgi(t(1),flo)b > 0

146

for all t(l) E Mi’ i = 1,2. Now
_ __ _ +
M1 - {x E L: g1(x,flo) - 0} — Gv
o
and
M2 = {x E J: g2(x,flo) = O} = Gv .
o
If M1 = M2 = ¢, then vO satisfies condition H. If
. i i+l i
M1 U M2 # ¢, then Dgi(t( ),mo)b = (-1) v(t( ),b) and by

choosing 6 sufficiently small we obtain v(x,9,IO + 6b)
satisfying condition H as follows. Without loss of generality,

assume M1 ¥ ¢. Now by the compactness of M there is an

1

open set U containing M1 such that for all x E U

v(x,b) > 0.

And again by the compactness of L ~ U there is a positive

number > 0 such that

61
V(X,9.IO) - L(X) 2 61

for all x E L N U. Then for 61 > 0 such that 61 3 ED

where H = “v(x,b)“, we have
v(x,9,[O + 61b) > L(x) for all x E L.
Similarly choosing 62 suitably small, we have
v(x,”b + 62b) < p(X) for all x E J,

thus 6 5 min (61,62) implies v(x,91O + 6b) satisfies con-

dition H.

 

147

If condition H is satisfied by v(x,wl) and for

Rn bi M #= 1 - 91 Th
m0 6 , 1 U 2 ¢, et b — m1 0' en

(1)

Dgi(t ,flo)b > 0

(i) . _
for all t E Mi’ 1 — 1,2. I

Remark 3: V1 1 is an equibasis system.
3

 

Proof: Suppose X = (e,M) is extremal for v(x,flo) 6 V1 1,

and that there exists Dv(x,mo)b E {a 1[MO] such that
’

min e(x)Dv(x,flO)b > O.
XEM

Now Dv(x,mo)b E i& 1[flo] implies
3

(1) (1)

r(l) ,mo)b 2 o for all t e M

V( ,b) = D81(t

1’
and

(2) (2) (2)

-v(t ,b) = Dg2(t ,mo)b 2 O for all t E M

2.
But then for some sufficiently small 6,

V(x,flo + 6b) = V(X.flo) + 6V(X.b) E V1,1

and Dv(x,mo)6b e {a 1[m0] with

min e(X)Dv(x,Mo)6b > O
xEM

implies
min e(X)(V(X.m + 6b) - V(X.m )) > 0
o o
xEM
Which is a contradiction.
Suppose X is extremal for O E {a 1[MO] and

there exists b E Rn such that v(x,b) 6 V1 1 and

 

148

min e(x)(v(x,b) - v(x,Mo)) > O.
xEM

Since v(x,b) - v(x,m0) = Dv(x,flo)(b - mo) and

Dg1(t(1),flo)(b - mo) = v(t(1),b) - v(t(1),mo) 2 O for all
C(l) E M1.
-Dg2(t(2),flo)(b - M0) = v(t(2),b) - v(t(2),flo) s O for all
(2)
t E M2:

Dv(x.9lo)(b - 910) e 1.1 IMO]. But

min e(x)Dv(x,mo)(b - $0) > 0
XEM

is a contradiction.

 

We also have the Signature z[f - v0] = (e,M[f — v01) where

M[f - v0] = {x E K: If(x) - v0(x)| = Hf - VOHK} = Ev
and

f(x)-vo(x)

e(X) = W for X E MEf " V0].

Then Theorem 3.6 is equivalent to Theorem 1.2 and Theorem

3.10 is equivalent to Theorem 1.3.

+
For Chapter II, let E = RS t, and for
s+t
fl = (a1,...,as, b1,...,bt) E R ,
s
iElaiui(X)
F(ala'00388: b1:'°°:bt) = r(x,91) = t
Z b.W.(X)
j=1 J
t
for m such that Z bjwj(x) > O for all x E S = J U K U L.
i=1

The side conditions for the case

 

149

L(x) < p(x) for all x E J = L

 

 

are:
g1(X:QI) = r(x,21) ' L(X) for X E L:
82 (X3941) = IL(X) " r(X,9.I) for X E J. r
_ o 0 o o o =
Then for mo (al,az,...,as, b1"°°’bt)’ b (a1,a2,...,as,
b1,b2,...,bt), r(x,mo) = ro(x),
Dflwon .=— o, '
1 ’ S t
Dr<x,mo)b = t (931891“) - r0<x).§lbjwj<x) ,
2: b‘.’w.(x> 1" J“
j=1 J J
Dg1(x,m0)b = Dr(X.910)b.

D82 (X $910)!) = "Dr (X 9m0)b '

Then the side conditions are (R1)-regular as before and (R2)-
regular at no if and only if there exists ¢ E P + raQ such

that

$(x) >0 for X 6M1

and

¢(x) < O for x E M2.

Remark 4: V1 1 is again an equibasis system.
,

Proof: Suppose X is extremal for r(x,m0) E V1 1 and there

exists b E E such that Dr(x,m0)b E {a leo] and
9

min e(x)Dr(x,MO)b > O.
XEM

150

6 can be chosen sufficiently small so that

 

 

 

S
2 (a? + 5a.)h.<x>
1=1 1 l 1.
t = r6(x) 6 v1 1. Then
2 (b? - 6b.)W.(X)
j=1 J J J
6 S
r5(x) - ro(x) = t (.E ai”i(x) :-
o 1—1
2 (b. - 6b.)wj(X)
j=1 J J
t
+ r b.W.(X))
is Such that ‘

min e(x)(r (x) - r (x)) > 0
xEM 6 O

which is a contradiction.
Conversely if 2 is extremal for O E ii,1[flo],
and there exists r(x,b) E V1 1 such that
3
min e(X)(r(X.b) - f(x,” )) > 0.
o
xEM

then

t
a.u.(X) - r 2
1 1 1 o j=l

_ 1 or
r(x,b) ' r(xamo) t L

2 b w (x)
j=1 j j

bjo(x)] .

"Mm

i

Thus

min e(x)Dr(x,mo>b > 0.
xEM

which is a contradiction.

Then Theorem 3.6 is equivalent to Theorem 2.3 and

Theorem 3.10 is equivalent to Theorem 2.4.

151

For the case L(x) S p(x), the side conditions are:

s t
{1(M) = ,2 aiwi(X1) a10 .Eb j wj(x1 )’
1=1 J= -l
S t t
f (m) = 2 a.w!(x ) - a 2 b v (x ) - a z b,vf(x ),
2 . i=1 1 1 11j___1 JJ 10 j=1 J J 1
° m -l
s (ml-1) 1 ml-l t (m -1-1)
f (fl) = 2 a.w. (x ) - 2 {(1)8 jzb (x1)},
m1 i=1 1 1 l 1=1 1 ,1- l _1 j vj
S t
f (91)=2a.W.(X)-a ZbV.2(X),
m1+1 1=1 1 l 2 20 j—1 j]
s (m -1) mk-1 mk"1 t (mk'l'X)
— _ 1
f k (u) — )3 a1": (xk) 23 {( x mkm1 E ijj (xkh.
1-1 )V-l J 1
2 m.
. 1
1=1

g1(xam) r(xau) ' L(X) for X E L:

g2(x,m) = p(x) - r(x,M) for x E J.

Remark 5: These Side conditions are (R1)-regular.

s+t
Let b = (c1,...,cs, dl"°°’dt) E R and

Dfi(fl)b = 0 for i = 1,...,m =

"MK"
3

Set w(t) = m + tb, for t E [0,1]. Since the fi's are

linear in m, Dfi(fl)b = fi(b)’ and Dfi(M)b = 0 for
i = 1,...,m implies
fi(m(t)) = fi(m) + t fi(b) = 0

for all t E [0,1]. Also M'(0) = b.

 

152

+t
Remark 6: (R2)-regularity is impossible, since if b C RS

is such that

n
c>
H3
0
H
H

n
H

U
3

Dfitmo)b

then fi(u)b = O for i 1,...,m. That is

.9b) = X. = X. for ' = 1,...,k,
r(xJ L( J) u( J) J

and thus
Dg1(xj,fl)b = Dg2(xj,fl)b = 0 for j = 1,...,k.

However if we redefine Q1 to be L ~ U and Q2 to be
J ~ U where U is any open set containing T, then (R2)-
regularity follows from the existence of an element ® E Mr

(where r = r(x,m)) such that

¢(x) > 0 for all x E L ~ T

and

¢(x) < 0 for all x E J ~ T.

V1 1 is again an equibasis system.
’

The results concerning uniqueness obtained by
Hoffmann deal with uniqueness of best restricted approxima-
tion to every continuous function f.

Example 1 of Chapter I shows that Hoffmann's unique-
ness theorem cannot apply to the work of Chapters I and II.
In this example we let f(x) - 1 - x2, K = [0,1], J = [0,1],
L = [0,1] with p(x) = 0, L(x) = -1. Then we considered

best restricted approximations to f from

 

153

2
V1 1 = {8x2 +'bx + c: L(x) s ax + bx + c g p(x) for all
3

x E [0,1] where a, b, c are real numbers}.

Now v1(x) E 0 E V1 1 and v1 is a best restricted approx-
3

. . 2 . . .
imation also v2(x) = -% x is a best restricted approx1ma-

tion. Thus f does not have a unique best restricted approx-
imation.

The results of Chapters I and II beyond the char-
acterization theorems do not follow from the material pre-

sented in Chapter III.

 

BIBLIOGRAPHY

 

10.

11.

12.

BIBLIOGRAPHY

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E.W. Cheney, Introduction to Approximation Theory,
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E.W. Cheney and H.L. Loeb, Generalized Rational Approx-
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N. Dunford and J.T. Schwartz, Linear Operators, Part I:
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, Nichtlineare Tschebyscheff-Approximation mit
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(1970) pp. 383-393.

S. Karlin and W.J. Studden, Tchebyscheff Systems: With
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J.J. Kelly and I. Namioka, Linear Topological Spaces,
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A.N. Kolmogorov, A Remark on the Polynomial of P.L.
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155

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