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" LIBRARY
Michigan State
University

 
      

This is to certify that the
thesis entitled
TCHEBYCHEFF APPROXIMATION
BY RECIPROCALS OF POLYNOMIALS 0N [0,00)

presented by

Daryl Myron Brink

has been accepted towards fulfillment
of the requirements for

Ph.D. Mathematics

degree in

4244/ 0- 944%

Major profegér

 

. Date‘irgugust 23, 1972

0-7639

 

 

 

ABSTRACT

TCHEBYCHEFF APPROXIMATION
BY RECIPROCALS OF POLYNOMIALS ON [0,”)

By

Daryl Myron Brink

In Chapter I we consider the following approximation problem. We
let n be a fixed positive integer and use C:[O,w) to denote the
set of all positive continuous functions on [0,”) which vanish at

infinity. We define

Rn = {gt-pent“ new on {0.00)}

and for a given f e c:[o,w) we search for an element -%* from Rn

 

such that
sup f(x) — -—l;- - inf sup f(x) - 1' .
p*(X) 1 p(X)
xs[0,w) ScRn xe[0,w)

In this setting we obtain existence, characterization and uniqueness
theorems. Existence follows along somewhat standard lines. The
characterization theorem features a two-part alternation condition
which states that ‘fi- is a best approximation to f if and only if
either f - %- has an alternating set consisting of at least n + 2
points, or f - %‘ has an alternating set consisting of exactly n + 1
points, 3p §.n - 1, and f -‘%- is positive at the largest point
of the alternating set. Uniqueness is then obtained.

In Chapter II we consider approximation with osculatory interpolation

in a setting similar to that of Chapter I. Here we obtain a character-

Daryl Myron Brink

ization theorem similar to that of Chapter I. Although existence fails
we have uniqueness of best approximations. If %‘ is best with
3p = n, we obtain a characterization in terms of zero in the convex
hull of a certain set.

In Chapter III we consider strong uniqueness and continuity of
the best approximation operator in the above setting and obtain an
existence theorem for approximation with osculatory interpolation for

large n.

TCHEBYCHEFF APPROXIMATION

BY RECIPROCALS OF POLYNOMIALS ON [0,”)

By

Daryl Myron Brink

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1972

4‘33)

ACKNOWLEDGMENTS

I wish to express my sincere appreciation to Professor Gerald D.
Taylor for his guidance in the preparation of this paper. Without
his concise insight and careful suggestions this would not have been
possible.

I also wish to thank Professor T. L. McCoy for his comments.

ii

Chapter

I

II

III

TABLE OF CONTENTS

INTRODUCTION O...OOOOOOOOOOOOOOOOOOOOO 000000 0...... 0000000

TCHEBYCHEFF APPROXIMATIONS WITH RATIONAL FUNCTIONS

OF THE FORM % 0N [0’m) OOOOOOOOOOOIOO0.00.0.0... .......
Section 1: Introduction ........................ .........
Section 2: Existence Theorem ............................
Section 3: Characterization of Best Approximations ......
Section 4: Uniqueness of Best Approximations ............

TCHEBYCHEFF APPROXIMATION BY RATIONAL FUNCTIONS OF

THE FORM % 0N [0,oo) WITH OSCULATORY INTERPOLATION . . ..

section 1: IntrOduction OOOOOOOIOOOOOOOOOOOOOOO0000......
Section 2: Characterization Theorems ............. ..... ..
Section 3: Uniqueness of Best Approximation .............

CONTINUITY OF THE BEST APPROXIMATION OPERATOR ............
Section 1: Introduction .................................
Section 2: Continuity of the Best Approximation Operator

for RI!OOOOOOOOOOOOOOQOOOOO0...............O.
Section 3: Results for Kn(f) ...........................

BIBLIMRAPHY 0......OOOOOOOOOOOOOOOOOOO ...... 00...... .....

iii

Page

POI-‘00

23

23
24
46
49
49

49
62

INTRODUCTION

This paper will consider the approximation of certain positive
continuous functions defined on [0,m) by rational functions of the
form 'fi- where p is a polynomial of degree not greater than some
fixed integer n. Existence, characterization, uniqueness and
continuity of the best approximation operator will be examined. The

error in approximating f by %* will be measured throughout by

(0.1) "f - 5| = sup f(x) - 5%)?)- .
x€[0.°°)

Thus a best approximation to a given f is an element %} from the

prescribed class such that

(0.2) "f _ it?“ = inf "f — a] .
E

In Chapter I, we assume that f vanishes at infinity and that p
is positive. In this setting, existence and uniqueness theorems
paralleling those found in the classical theory of rational Tchebycheff
approximation [1], [2],[1o], [14], [15] are obtained. The characteri—
zation theorem, which involves conditions on the sign alternation of the
error function, has non-standard form.

In Chapter II, we further restrict the functions and the approx-
imants and consider the problem of approximating an s-continuously
differentiable f by a k-point osculating rational function of the
form '%- (precise definitions will be given later). In this setting,

we do not necessarily have existence of best approximations; but

uniqueness and characterization theorems similar to those in Chapter I
are obtained. The results here are generalizationsto the infinite
interval of work by A. L. Perrie [13] and H. L. Loeb, D. G. Moursund,
L. L. Schumaker and G. D. Taylor [9]..

Chapter III consists of a discussion of continuity of the best
approximationoperator for the above approximation problems, along with

an existence theorem for large n."

CHAPTER I

TCHEBYCHEFF APPROXIMATIONS WITH RATIONAL FUNCTIONS

OF THE FORM % ON [0,”)

Section 1: Introduction

In this chapter we wish to consider the following approximation
problem. Let C:[O,w) be the set of all positive continuous functions

on [0,w) which vanish at m. Define

sup |f(x)|.
xe[0,m)

llfll

Then, given an integer n, we wish to approximate f by a function

of the form %- where

with ai real. We will use Sp to denote the degree of the poly—

nomial p.

Let

Rn = {filapim P(X)>0 on [0.W)}

denote the family of rational functions from which we wish to approximate

f. Thus we search for an element %% from Rn such that

"2.51,” = neg—u.

-l-sR
p n

Note that, since f e C:[O,W), the constant c* E %-Hffl is
always a candidate for a best approximation; moreover, we have

u: - den = § llfll . Thus
(1.1) "f “11;," 1 %llfll .

Hence p* has no zeros, and so there is no loss of generality in
requiring p > 0 on [0,m) for fi‘e’Rn. However, note that 0 ¢ Rn'
The question of whether such an element %} 6 RD can actually be

found is answered in Section 2.

Section 2: Existence Theorem

We begin with an important lemma which will have applications

throughout this paper.

Lemma 1.1: Let { A-} be a sequence of elements from Rn' Suppose
k

there exist constants M1 and M2 such that

“Ml-ill}: "1M2”

for all k. Then there exists a subsequence { l-} of { 3-} which
pkj Pk

converges uniformly to %- on any closed interval, with %'8 R .

Proof: Since {" %- H} is a bounded infinite sequence, there is a
k

subsequence { %‘} such that {:"‘%’ "} converges to some real number
k k
r r

L as kr + m. For simplicity we will denote this subsequence by { %’}.
k

We write
3.-.?1.
Pk qk
“ 1 n2
where q (x) = Z a x with Z a = l and c is a constant.
k . i,k i,k k
i=0 i=0
n 1
To do this, suppose p (x) = E b x and take
k i,k
i=0
n
2 _ 2
ck ‘ 1:0 bi,k°
n+1
Now {(a ,...,a )} is a bounded infinite sequence in E
o,k n,k
and therefore must have a subsequence {(a ,...,a )} which
o,k n,k
J J
converges to some (ao,...,an). Define
“ 1
q (x) = Z a x .
kJ 1_0 i,kJ
n 1 n
Then if q(x) = 2 six , since 2 Ia1 - a1 k I + 0 as kj + m,
i=0 i=0 ’ j

using a result from Natanson ([11], p. 23) we find that { qk} converges
J

uniformly to q on any closed interval.

For simplicity, we will write {qj} for { qk}. Consider
J

the associated sequence {c }. Since

3

“2g:

 

2

we have

c
q(X) - 2
J
for any x. Hence
c. < M 1
IJI _ zlqj<>|
n
f. M 2 la l
2 1=o 1’3
_<_ M2 (n+1)
“ 2
since 2 a . = 1 implies that Is I < 1. Thus {c.} is a
i=0 19.] 1:j _ J

bounded sequence and therefore has a subsequence {cj } which
t

converges to a real number c.

We wish to show c > 0. If C3 + 0 then the coefficients of
t

p, must approach zero, which would force
t

to become arbitrarily

 

 

 

I}.
P
jt

large. Hence 0 < c < w, and consequently q has no zeros. Thus

we let = in
P

2.
q
Let a be a real number and suppose x e [0,o]. Then

r s

1 l J 1

1300 - 15>j (X) i
t

 

‘ Pj (X) - 900
min Ip(x)pj (X)| t

xe[0,a] t J

 

 

Since pj + p uniformly on [0,a] with p > O, > 0 on [Osa]
t

pjt

it follows that

uniformly on [0,a]. Thus { fi‘} is the desired subsequencea I
J
t

With the aid of lemma 1.1 we can prove the existence theorem for

Theorem 1.1: To each function f e G:[O,®) there corresponds at least

one best approximation %; from the class Rn'

Proof: Fix f e G:[O,w) and let

E = inf "f—-1-"
1 P
-eR.
p n
Then there exists a sequence { %-} from Rn such that the sequence
k

{Hf - %‘"} + E as k + w. It follows from (1.1) that there is no
k

loss of generality in requiring that

Inf-2k" : 2“"

for all k. Thus it follows that
l l 3
2 Hill 5, " pk " _<_ 2 ufu .

We wish to construct an element '%* from Rn such that

"f - fit" = E. By lemma 1.1, we have a subsequence { %-} which
k
j

R .
n

l
converges uniformly on any closed interval to an element PT 8 Rn.

Let X 8 [0,”). Then x 8 [0,0] for some a, and we have

 

 

 

 

 

 

 

 

 

 

l . 1
f(x) — 3:7;7- 5_ max f(x) - 5:7;7
Xe[o,a]
< max f(x) " “—1—"- + max 1 - 1
“ Pk (1:) pk (X) 13* (X)
xe[0,a] j XE[0:G] j
l __1__ ___1___
f. ”f — Pk + max Pk (x) _ p*(x)
J xe[0,a] j

+ E we have,

 

 

3':
k P k

Since %- +~l uniformly on [0,o] and “f --%
j

j

by taking limits on the right hand side of the above inequality, that
l
f x —-—--— < E .
|() P*(X)| -

Hence it follows that

and consequently, %* is a best approximation to f.|

We remark that the proof given above parallels the proof of the
well-known existence theorem for approximation by rational functions
on a closed interval. The technical difficulties which arise as a
consequence of the use of an infinite interval are somewhat simplified
by the assumptions on the function f and the use of a smaller class of

rational functions.

Section 3: Characterization of Best Approximations

Our objective here will be to characterize best approximations by
means of the sign alternations in the error curve f - r where

r e Rn' In order to do so, we will first define the following sets:

(1.2) X(r) = { x [f(x) — r(x)| = Hf - r" } is called the set

of extreme points for f - r.

(1.3) A set of N distinct points 0 :_x1 < x2 <...< KN < m

is called an alternating set for the error f - r if

x a X(r), j = 1,...,N

J

f(xj) — r(xj) = - [f(x ) — r(xj+l)], j = l,...,N—1.

j+l

Now for f e C[a,b] with a,b real, it is well-known that r is

a best approximation to f from the set
Rm[a b] = { R- p e H q a H (x) > 0 for all x e [a b] }
n s q m: n: q y
if and only if f - r has an alternating set consisting of
N = 2 + max (n + 3p, m + aq)

points [15, p. 122]. Since the set Rn is contained in R:[0,m) we might
expect a characterization theorem in terms of an alternating set
consisting of N = n + 2 points. Achieser [1] gives such a theorem

for infinite intervals in the case m z_n with additional conditions.

This theorem, however, does not handle the case under consideration here.

10

The following example shows that some modifications of the usual
theorem will have to be made for characterizing best approximations to

+
f e Co[0,w) from Rn.

l

- _1_ _
Example 1.1. We will consider the approximation P(X) - x + l

to f e c:[0,m) where f is similar to the function defined as shown,

and prove %- is a best approximation to E from R2.

 

 

We assume f is constructed so that

f(0) =

f(1) =

f(2) =

'_l
N[u b4H blw

l l l l 1
- _. = _ _. = _. _ _. _. 2.
With X( p ) {0,1,2}, "f P" 4, and f P < 4 for x >

So if % e R2 is such that ”f - i“ < “f - %“ we must have

ll

 

 

 

 

 

1 l 1
f(x) q(x) 4 V x e X( p )
and
l l 1
‘max {0, f(x) — 4} < q(x) < f(x) + 4 .
Then at x = 0, we must have
1 5 l l
‘ 4 < 4 ' q(O) < 4
which implies that
%< q(O) < 1.
At x = l, we must have
1 l 1. l
4 < 4 q(l) < 4
from which it follows that
2 < q(l).
At x = 2, we need
1 7 1 1
‘ 4 < 12 ‘ q(2) < 4
which leads to the inequality
%< M) < 3.

Thus if q(x) = ax + b we must have

-% < b < 1
2 < a + b
-g-< 2a+b < 3.

But b < 1 implies a > 1, and hence a + (a + b) > 1 + 2 = 3, which
cannot happen. So q(x) must be of the form ax2 + bx + c. Thus

we have

12

uqox n: uflh>
A
m
+
a‘
+
O

In order that -% e Rh, we must also have a > 0. Now c < 1 implies

a + b > 1. So
4a + 2b + c = (a + b + c) + (a + b) + 2a > 2 + l + 0 = 3 .

Thus no such q exists. Hence ;%I is a best approximation to f

from R2, with an alternating set consisting of only three points.

Finally, we note that i is also a best approximation to f from R1.
Since the best constant approximation to f s G:[0,w) is

c* E %~"f" , we will assume n :_l in the characterization theorem.

In this case no best approximation can be constant. To see this, let

c* be the best constant approximation to f from Rn with n :_1. We

 

will show it is possible to choose a and b so that e R and

l
ax+b n
llf — fill < llf - c4" = 29:
Since f(x) > 0 for all real x, we must have
f(x) — c* > 0
for all real x e X(c*) and

lim f(x) - c* = — Hf — c*H .

x—Mo

Thus there exists a real number a such that f(x) < c* on (a,W)

13

and f(a) = c*. Note that [0,a) contains X(c*) ” {W}.

Now let

25 = min f(x).
xe[0,a]
Let
1
b = 2* - ac

and choose a > 0 so small that

1 l
c*+€ c*

 

Then for x s [0,a), we have

1
ax + b = a(x — a) + 2* <-—

c*
and
l
ax + b = ax + —- - as
c*
> -l - as
_ c*
> _1_
C*+e
Thus
c* < -l- < c* + s
ax+b
for x e [0,a). But
1
____ * =
ad+b c f(a)

and hence it follows that

14

1

 

 

_ *
max f(x) ax+b < c .
xe[0,o]
1
Since x+b is decreasing on [0,”) we also have
max f(x) - -l- < c*
l ax+b ’
xe[a,w)

and hence

1
"f ‘34:" < C"

This leads us to suggest the following theorem for characterization

of best approximations.

Theorem 1.2: Let f e c:[o,m) and n z_l. Then %- is a best

approximation to f from the set Rn if and only if either

(1.4) f - %- has an alternating set consisting of at least
n + 2 points,
or
(1.5) f —'% has an alternating set consisting of n + 1 points,

3p :_n - 1 and f — % is positive at the largest point

of the alternating set.

Proof: We will first show that if we have either (1.4) or (1.5) then

.l is best.
P
Suppose (1.4) holds. Then % cannot be constant. Denote the

l m
a ternating set by 0 :_x1 < x2 <...< xn+2 < so that

15

1 _ _ 1 =
f(xj) - EYEEII — "f EH , j l,...,n+2.

If there is an element i-e R.n such that
1 l
1.6 f - - f --
< > u qu < n pu

we must have that

l>

1 l l
.._ _. = f _ _. _ f _
q P ( P) ( q

alternates in sign on the set x < x < ... < and hence has at

1 2 xn+2’

least n + l zeros. But

with pq > 0, and so p - q has at least n + l zeros. Since p — q
is a polynomial of degree at most n, we must have p - q E 0. Thus
p E q and (1.6) cannot hold.

Now assume we have (1.5). Again %- cannot be constant, so let the

< x < ... < x < m, and suppose there is

alternating set be 0 :_x n+1

l 2
an element i-e Rn such that (1.6) holds. By repeating the above
argument, we guarantee that there are at least n zeros for p - q.
80 if aq :_n — l, we find that q E p which again contradicts the
assumption (1.6).

So we have only to consider the case where aq = n. Since we

assume (1.6) holds, and

l

f(xn+l) _ p(xn+l)

> 0

we have

l6

o<(f--1-)-(f—-1-)=_1__l=a;_a
p q p q pq

Hence p - q > 0 at x Since -l s R , q must have

at x n+l° q n

n+l°
a positive leading coefficient. Here 8q > 3p and so p - q has a
negative leading coefficient. Thus p - q + -w as x + m, which

implies that p - q has a zero x* > x So p — q has n + l

n+l°
zeros, and again we must have p E q. This contradicts the assumption
(1.6) and therefore -%- is a best approximation to f.

To complete the proof of the theorem we will show that if ‘%
is best and (1.4) does not hold, then we must have (1.5). So assume
there are N < n + 2 points in the alternating set. Then if 8p = n,

l
we can construct a better approximation '3} as follows:

Begin in the usual way by constructing N subintervals

[0951]: [£1952]:°°°9[gN_19“)

in such a way that in each of them in turn one of the following two

inequalities is satisfied:

 

 

l
(1.7) - E :_f(x) - p(x) < E - a
(1.8) -E+a<f(x)— 1 <13
13(24)-
where E = “f - fin and a is some positive real number.

Following Achieser [1, p. 55] we consider the function
¢(X) = (x - £1)(x - £2) ... (x - €N_1)

and write ¢(X) = kp(x) - ¢(x) where 3¢ = n and k is a constant.
Then set

c l - bk

q(X) "- p(x) - b¢(x) '

We will show that there exists b* > 0 such that fi-e Rn for all

17
b with [b] §_b*. Then, since

__ c = _ 1 b¢£x>
f (X) q (X) f (X) P (x) + p (x) [p (X) ' 13¢ (1‘)]

and b¢ changes sign at $1,...,£ if we show that

N-l’
P(X)[P(X) - b¢(x)] > O on [0,m) and that

¢(x)
P(X)[P(X) - b¢(X)]

is bounded, it will follow by taking b small enough with the appro-
priate sign that E- is a better approximation to f than fin

Let b0 > o be such that 1 — bk > o for all b with |b| :bo.
Let an denote the leading coefficient of p. Then an > 0 so there

exists b1 > 0 such that if lb] §.b p - b¢ has a positive leading

1’

coefficient.
Now observe that there exists A > 0 such that p(x) :_21 on
[0,W). Then it is easy to show that there exists 8 > 0 such that

p - b¢ :1 > 0 on (8.”) for all b with |b| _<_b1.
Now there exists b2 > 0 such that |b¢(x)| §_A for all x s [0,8]

whenever lb] §_b2, and thus p(x) - b¢(x) :_A on [0,8] whenever

lbl _<_b2.

Finally let b* = min {bo, b b Then if lb] :_b* we have

1’ 2}.

E- in R , and
q n

b[k - 93324
b¢(x) - 1p(x)
p(x)[p(x) - b¢(X)] [p(X) - b¢(x)]

approaches zero as x + m. Now we show that by taking b small enough
with appropriate sign we can make fi- a better approximation than -% .

For x such that (1.7) holds, we have

18

-E§_f(x)--—-c <E+ ”(’9

b¢(x) _
q(X) p(X)[p(x) - b¢(X)]

P(x)[p(x) - b¢(x)l

0.

so choose b such that b¢ > 0 and

b¢(x)
P(x)[P(X) - b¢(X)l

Then b¢ < 0 where (1.8) holds, and so if b is so small that we

also have

bQ(x) 1 > _
p(X)[p(X) - b¢(X)] a

where (1.8) holds then -§’ is a better approximation to f than %u
Thus we have shown that if ‘fi- is best, and (1.4) does not hold, we
must have 3p §_n - 1.
Now if we have N' < n + 1 points in the alternating set, we
can again construct a better approximation than -%. If 3p = n - l,
we proceed exactly as above. If 3p < n - l we use a slightly modified

but simpler approach.
Mw=(x-fiHx-Q)u.@-€wd)
has degree at most n - 1. Choose a = l or e = -1 so that

sgn e<l>(xi) = -sgn (f(xi) - ML?) .

Then if the leading coefficient on s¢ is positive, we define

l
* = I— '
¢ e¢. If not, we select EN.) xNglsuch that X(p) c: [0,5N) and set
¢*(x) = e¢(X) (gNi- X).

So in either case, be* has a positive leading coefficient for any b > O.

19

Now there exists 8 > 0 such that
2
p(X) + b¢*(X) > P(X) > E
and [f(x)] < %- on (8,m). Thus for any b > 0 we have

1
p(x) + b¢*(x)

1
f(x) - p(x) “mm _<_ |f(x)|

 

+

 

 

E
+ 2 - E.

A
bond

on (8,w). As before we can select b so that

1
f(x) ‘ p(x) + bd>*(x)

 

on [0,8]; hence g; is not best.
So if (1.4) does not hold, we have exactly n + 1 points in the

alternating set.

Finally we must show that if is best and (1.4) does not hold,

we have
1
f(x ) - -—-——- > 0.
n+1 p(xn+l)
where xn+1 is the largest point of the alternating set. 80 assume
1
f(x ) '——““‘ < 0.
n+1 p(xn+1)

We know f - %- changes sign at at least n points 21 with

X (Z

i i = 1,...,n. Let

1 < x1+1’

q(X) = 6

(x-Z)
1 1

l

":16

20
where 5 is either +1 or -1 and is chosen so that

sgn q(xi) = -sgn (f(xi) — 37%;3->.

Then q(xn+l) > 0 and q(x) + m as x + m since all zeros of q

Thus there exists a > 0 so that -l-- is
o p + sq

in Rn for any a with 0 < e < co, and so we will construct an

appear before xn+1.
element of this form which is a better approximation to f than -l.
E 1 E
Choose 8 > 0 so that both f(x) < 2 and p(x) < 2 on (8,m).
Then [0,8] contains the alternating set, and p + eq > p on (3,”)

for any a > 0, so it follows that

-_1__
p+€q

on (8,w) for any a > 0.

We must show that e > 0 can be chosen so that

1

f_p+€q

on [0,8]. To do so, we note that there exists some a > 0 such that

(1.7) holds on (zn,8), (Zn—2’ z 1), ... and (1.8) holds on

n—
(Zn—1’ zn), (Zn—3’ zn_2), ... .

On the intervals where (1.7) is satisfied, q(x) > 0 so for

e > 0, p + sq > p and thus

We need to choose a > 0 small enough so that

21

This will be the case if

.2
2

.2

2 E).

1 > pf + p - pE + €q(f +

Since f - %?< E - a we have 1 > pf + pa - pE. So we need

eq(f +%- E).

'd
5H9
v

Since p(x) {_A for all x, and both q and f assume their maximum

value on any closed interval, there exists some 81 > 0 such that the

above inequality holds when 0 < s < 61. Thus we have

1 a
— -—————< -—
E < f p | sq E 2

on (zn.B), (Zn-2’ zn-l)’ ... when a > 0 is less than min {60,61}.

Similarly we find that there exists 82 > 0 such that

a 1
2 - E < f - p + eq < E

on (Zn-1’ zn), (Zn-3’ zn_2), ... when a > 0 is less than
min {80,82}. So if we choose 2* with 0 < 6* < min {60,81,82}, we

have

1 l
“f - p + 8*q" < If -‘3“

which contradicts the fact that -%- is best. Hence we must have

)--—L——— > o.l

f(
p(xn+1)

xn+1

Section 4: Uniqueness of Best Approximations

Retaining the setting of the previous sections, we obtain the

22

following uniqueness theorem.

Theorem 1.3: If %- is a best approximation to f s C:[0,w) from Rn,

 

then it is unique; that is, if :11- s Rn and in} %, then

1 l
f - — f - — o
u qu > u ,II

This theorem is an immediate consequence of theorem 2.3 in Chapter II,

and hence will not be proved here.

CHAPTER II

TCHEBYCHEFF APPROXIMATION BY RATIONAL FUNCTIONS

OF THE FORMI %' ON [0,”) WITH OSCULATORY INTERPOLATION

Section 1: Introduction

The problem we wish to consider in this chapter is that of
approximating a function f by a k-point osculating function of the
form 1%“ In order to be more specific, we retain the notation of
Chapter I, and introduce some additional terminology.

Let {y1,...,yk} be a fixed set of k points from [0,«) and

{m .,mk} a set of positive integers. Set

1’0.

and

s = max {m - l}.

i

i

In addition, we will assume that m < n + l and use Cs[0,w) to denote
the set of functions from c:[0,w) which have at least 8 continuous
derivatives.

Recall that a function f e C8[O,m) is said to have a zero of
order v 5. s at 2 if f(z) = ... = f(V'1)(z) = o, f(“)(z) # 0.

We will adopt essentially the same approach as that of Perrie [13],

and hence for a given f e C8[0,m), we define the set Kn(f) by

_1_ (J) _ (J) g _. =
(p) (Y1)‘f (yi), j o,...,m11, 1 1,...,k}

 

1
Kn(f) = {'5 8 RH

where f(yi) # f(yj) for some 1,1.
23

24

Then we will be interested in finding a best approximation to f from
the set Kn(f)° Such a best approximation may not exist. Gilormini [ 6]
claimed existence for rational approximation with interpolation on a
compact interval, but Loeb [53] published a short counterexample to this
claim. Here we will postpone further discussion of the existence
question until later, when we will obtain results in special cases, and
turn instead to characterization of best approximations.

An alternation theorem similar to that of Chapter I can be obtained,
with the modification that, roughly speaking, the order of interpolation
at the point yi may "use up" a certain number of alternations of the
error curve. Theorems of this type have been given in [9] by Loeb,
Moursund, Schumaker and Taylor in the case where f s C(X), X a compact
subset of [a,b], with approximants from a subset of an n-dimensional
extended Haar system of order v, and where the error of approximation
was measured with a generalized weight function. Their work generalized
results of Paszkowski [12] and Deutsch [5]. Perrie [13] gives similar
results for approximation to f e CS(X), X a compact subset of [a,b],

by functions from a subset of

{ 5‘] p s P, q a Q, q > 0 on X }
with P and Q finite dimensional subspaces of CS(X).

Section 2: Characterization Theorems

The following lemma due to Salzer [16] will be used. We assume

f e CS[0,w) is given with f(yi) i f(yj) for some i,j.

 

25

Lemma 2.1: At the points where q(yi) ¥ 0, the system

= 0,...,m —l
<§>m<yi> = f(j’oi), i

= 1,...,k
is equivalent to
— 0, ..,m -l
p”) oi) = (qf) (3) (yi). i
= 1,...,k.

We note that this equivalence does not require p or q to be
a polynomial or even a linear combination of given functions.
The analogue of theorem 1.2 for characterization of best approx—

imations can now be proved.

Theorem 2.1: Let f e CS[0,W) and n :_1. Then %- is a best

approximation to f from the set Kn(f) if and only if either

(2.1) there exist at least n — m + 2 consecutive points xi
such that
l l
a f — —————- = f — —
<> l(xi) W9] H p"
and

l
(b) Sgn [(f(xi) — p(xi) ) H(Xi)]
1+1 1
= (-l) sgn [( f(xl) - FEET) 110(1)]

for i = l,2,...,nrm+2 where

H(t) = (yl-t) (yk-t) .

26

or

(2.2) there exist exactly n - m + 1 consecutive points

such that (a) and (b) hold for i = 1,2,.

3p §_n — l and

l
(f(xn-m-l-l) -m) 11* > 0

with

mv

* = _

H H (xn—m+1 yv)
v69

where

x1

{yv}vsu is the set of interpolation points which are

= *3
larger than xn—m+l' (If {yv}vefl 0, let H _ 1).

Proof: We first show that if we have either (2.1) or (2.2) then

1

is best. Suppose we have (2.1) and that there exists -% e Kn(f) such

that
l 1
£-— f——.
n qu < u ,u
Since %-6 Kn(f)’ by applying lemma 2.1, we see that

j = 0,...,mi-l

l (j) = (j)
(p) (V1) f (Vi). i=1.....k

is equivalent to

j = 0,...,m -l
(j) _ 1 (j) i
p (y) — (-) (y).

i f i 1= 1,...,k.

But -l e K (f) and so
q n

27

= 0,...,mi-l

(j) _ (j)
P (y ) ~ q (y ),
i i 1= 1,...,k

which implies that p - q has a zero of order at least m at each

3
yj. Thus we have

zeros for p — q.

Now consider the relation

1_ _1___ = _1___1_
(f(x) ' p(x)) ”9" + (q(x) f9“) ‘1‘") (q(x) p(x)) “9"

ESX! " SEX!
( <pq><x) )H‘X)'

l 1
Since pq > 0 and lf(xi) - q(xi)| < |f(xi) - p(xi)| , we find that

( p(x) - q(x) ) H(x) alternates in sign at n — m + 2 points xi.
It is given that xi # yj for any i,j. If the sum of the mj's

such that yj 5 (xi, x1+l) is even, then

H(xi)H(x ) > 0

1+1

so that

[p(xi) - q(xi)llp(x ) - q(xi+l)] < 0 -

1+1

Thus p - q either has a zero in (xi, xi+1) ~ 121 {yi}, or

- +
p q has a zero of order mj l at some yj 1+1

argument is used if the sum is odd. Since there are n - m + 1 intervals

in (xi, x ). A similar

of the form (xi, x ), we have at least n - m + 1 additional zeros

1+1

28

for p - q. Thus p - q has at least n + l zeros, which implies p E q,
and this is a contradiction. Hence (2.1) implies -%- is best.

Now if (2.2) holds, and there exists 'i-s Kn(f) with

1 1
I14 -;n < Ilf an

we find that p - q has n zeros by arguing as above. So if
3Q §_n ' 1, we find p E q and again reach a contradiction. Thus, we

need only consider the case where Bq = n.

If H* E 1, then we have

-.1 ._ _.l; = 2.2.9
0 < (f p) (f q) pq

at and since pq > 0 we find that p — q > 0 at x

xn-uH-l ’ n-m-l-l °

But aq > 3p and q has a positive leading coefficient, and so

P ' q + 'w as x‘+ w. Thus p - q has a zero x* > x Hence

n-m+1°

we have n + l zeros for p - q, which again leads to a contradiction.

* _
Now suppose H is non trivial. At xndm+l we have

_.1 __ _.1 1 .24241
0 < [(f p) (f q)] H* ( pq )H*

_ *
and hence (p q)H > 0 at xn-m+l°
If the sum of the mv is odd, then H* < 0 and so p - q < O

at Let yk be the largest member of {yv}. Then p - q has

xn-m-i-l °
a zero of order at least m» at each yv in (xn-m+l’ yk]. If the

order of the zero yv is exactly mv for all v, and if there are no
other zeros of p - q in (xn-m+l’ yk] we must have p(x) - q(x) > 0

for x > yk. But then since p - q'+ -m as xr+ a, p - q must have

an additional zero x* > yk. In any case, we find p - q has n + l

29

zeros, which leads to a contradiction.
If the sum of the mv is even, then H* > 0 and so p - q > 0

at Arguing as above leads to a contradiction since if there

xn-m+l'
are no additional zeros in (xn-m+l’ yk], again we have p(x) — q(x) > 0
for x > yk.

Thus we have shown that either (2.1) or (2.2) implies that %
is a best approximation.
Suppose now that %- is best. We will show that if (2.1) does not

hold, we must have (2.2). So assume we have N :_n - m + 1 consecutive

points x such that (a) and (b) hold. If 8p = n, we can construct a

i
polynomial q such that 3—:qu s Kn(f) and
llf p + “—e‘l‘qll ||f ' ‘II

for some a > 0.
In order to do so, we first introduce the following sets:

(1) {21} is the set of zeros of f - %’.

(2) {wi} is used to denote the set X(— 1) ” {xi }-

(3) {Si} is the set of points which are different from {yi}

where sign changes of f - %- occur.

Notice that {$1} C2‘[zi} and {x1} n {wi} = ¢.

Immediately we make the following modifications to the sets above:

(1) If any of the sets contain an interval, we remove the interval
from the set and replace it by a single point from the

interval.

30

(2) Let c be chosen so large that X( %-) U {yi} is contained
in [0,c), with the additional restriction that if there

are any 81

least one of these 31 is included in [0,c).

which lie to the right of X(‘% ) U {yi}, at

(3) If there are consecutive wi with no other points from
{xi} U {Vi} U {21} between them, remove all but one of

these wi.

(4) Repeat (3) with wi replaced by 31 and {xi} U {yi} U {21}

replaced by {xi} U {yi} U ({zi} " {si})-
Using the same notation for the modified sets, we find that
{xi}, {yi}, {21} and {wi} are all finite.
We wish to construct a polynomial q(x) with behavior similar to

that shown below:

 

 

Let Ix = (a1,bi) be the largest open interval containing x
i

i

with all other points of {xi} U {yi} U {81} exterior to Ix . Then
1

between Ix and I there must be a yi or an s In fact, the
1

1+1 1

31

 

end points of [a1,bi] and [ai+l’ bi+l] belong to {yi} U {81}.
For convenience we let
E<x> = f(x) - 1
MI!)

and consider

Case 1: Suppose there are no points of interpolation yi in

[bi’ a1+1]. Then the only points from. {W3} which can be contained in

[b are those at which

1’ 31H]
sgn E(xi) = sgn E(wj).

Hence q(x) need not change sign on [a1, ai+1] and thus need have

no zeros in (a1, ai+1). So with this xi we associate a factor

(x - ai+l) and include this factor in q.

For Ix we include 6 = :_l in q so that
1

San E(x1) = ~83n q(xl);
thus q will be constructed so that
sgn E(xi) = -sgn q(xi), i - 1,...,n-m

in either case.

A typical situation for Case I is shown below.

 

 

 

 

 

 

32

Case II: Suppose we have at least one yi in [b Beginning

1’ a1+1]'
at bi’ we can list the points of the set

({yi} U {51} u {W1} > n [bi.ai+11

in increasing order. We need not consider any 91 before the smallest
yi in the list, since there can be no WE with

sgn E(x1) = —sgn E(wj)

before the smallest yi. (If there were such a “a, it would have
to be an alternation point.) In order for q to behave as required,

it may or may not have to change sign at the yi's and 31's, so

we need the following scheme to decide what type of factors to include

in q. We consider the yi's and 51's in the list starting at the

first yi.

(*) At yi, we either

m
(1) put a factor (x - yi) i into q

or

1111+].
(2) put a factor (x - yi) into q
according to the following instructions.

(1) If the next point is a w then m is even if

i’ i

sgn E(wi) = sgn E(xi), and m1 is odd if sgn E(wi) #
m
sgn E(xi). In either case we need only a factor (x — yi) i

in order to insure q has the correct sign here.

(ii) If the next point is x1+1 then m1 is even if

and m is odd otherwise. In

sgn E(xi) # sgn E(xi+l) 1

33

either case q will have the correct sign if we include a

1111+].
factor of the form (x - yi) .

m
(iii) If the next point is yi+1, put a factor of (x _ yi) i

into q and then repeat the analysis for y1+1 going back

to (*).

(iv) If the next point is an 81 we proceed to the following point

appears directly after 3 .

in the list unless a w or x i

1 1+1

If we have a w directly after 5 and the sum of the

1

previous m '9 associated with the yj's that occur from

3

xi up to this point is even, then we will have

i,

sgn E(w1) = sgn E(xi) and q has sign opposite that of
E(Wi) here. If the sum is odd, then sgn E(xi) ¥ sgn E(wi)
and again q already has sign opposite that of E(W1)°

Hence, q need not be altered here.

If x

1+1 then we may have to

appears directly after 31

include a factor (x - Si) in q.

Thus we proceed to y1+1, and repeat the procedure beginning again at

(*), if necessary, until we finish with a Thus we arrive at

i+1°

x1+1 and then continue by going back to Case I.

From the above analysis we conclude that the only time it is
m
necessary to include a factor beside those of the form (x - yi) 1 in

q is directly before x (1 ¥ 0). Letting q be the product of

i+l

all the factors (x - yi)mi and including only necessary additional

factors we find that

34

k
3q §_ 2 mi + N - 1 = m + N - 1
i=1
and q has opposite sign from E(x) at all points of X( %-).

Furthermore,
(p + eq>‘3’(xi> = p‘j’cx1> + <eq>(3)<x1>

. (no. 1 _. =
p (x1), j 0,...,m1 l, i 1,...,k.

Thus for a constant, we find that p + sq has the interpolation
properties specified for p.
Then as in the proof of theorem 1.2 we can show there exists a

1
such that p + eq 5 Kn(f) and

IF‘p+eJ "f'i

So if (2.1) does not hold, we must have 8p §_n - 1. Then if
N :_n - m we have aq §_n - l and again we can construct a better
approximation than %3

It remains to show that

1

(f(xn-nfil) - p(x

) ) H* > 0.
n~m+l

If H* E l, we proceed as in the proof of theorem 1.2. So assume that
H* is non-trivial, and that
l
H* < O.
>)

(f(xn-m+l) - p(xndm+1

Then if 2mV is even, H* > 0 and so f - %-< 0 at x Since

n-m+1'

35

sgn (f(x1)—;(—’1‘1—)) = —sgn q(xi)

we have q(xn_m+l) > 0. The only factors included in q after
mv
xn—m+l in the earlier analysis are those of the form (x — yv) ,

and hence q(x) > O for x > yk. Thus q has a positive leading

coefficient and, as in the proof of theorem 1.2, we can show that

there exists 5 > 0 so that S—:;;;- is a better approximation than %-.
* _ l. .
If Emv is odd, H < O, f p > 0 at xn_m+1 and q(xn-m+l) < 0

Then arguing as above, we see that q(x) > 0 for x > yk and again
we can find a > 0 so that IT:%:;I is a better approximation than %-. I

It is useful to obtain a slightly different characterization
theorem, which states that under certain conditions -% is a best
approximation to f if and only if zero is an element of the convex
hull of a certain set. In the case of generalized rational approximation,
a theorem of this type was proved by Cheney [2]. For ordinary
interpolation, Gilormini [6] gave a similar result. Perrie [l3] proves
such a theorem in the previously mentioned setting.

We will need some preliminary results. Recall f is a given

function from CS[0,w). Let fig e Kn(f) and define

1 1
3(34 = { 1 “3* | q 6 En. q(j)(yi) = (;)(j)(yi). j= 0,...,m1-1; 1 = 1,...,k}

Note that since f > O on [0,m) we must have q(yi) > 0 for all 1.

Lemma 2.2: S( %* ) is a subspace of { g} I q s Hn }

Proof: Let 1 - g} and 1 — g} be elements of S( %k ), and let

c be a real number. Then

36

C(1_%*)+(1__;1*) = 1_§LL*2_+_<1

p4
Now C(p - p*) + q s nn and
[C(9 ' p*) + q](j)(y1) = c ( p(j)(yi) - p*(j)(yi)I + q(j)(yi)
= ( %a(3)gyi> .
Thus c(l — fig) + (1 — fig) 6 s( %k ).l

Lemma 2.3: Dim S( fik ) = n — m + 1.

Proof: Let N = n — m + 1. Choose N distinct points x1,...,xN
which are different from the interpolation points y1,...,yk. For
each C, t = 1,...,N we will show that there exists gt a S( %k) such

that

o t i j
8t(xj) = th 1 t = j

for j = 1,...,N. So let h* be any element of S( %%), and write

* _
h* in the form 1372 .

For each t, there exists a polynomial qt of degree at most

n with the following properties:
(1) (j)( ) = 0 - o -1- 1 = 1 k
qt Y1 I j— ’oct,mi , ,..I,
(2) qt(xi) = p*(xi) - p(xi), 1 + l,...,t-l,t+1,...,N
<3) qt(xt) = -p(xt)

We set

37

g = 1_(p+q,_-).
t p*

Since p + qt 2 Hn and (p + qt)(J)(yi) = p(j)(yi) we have

 

l
gt 6 S( PE ) . In addition,

* - .
g(X)= p(xj) (p(xi)+qt(xj>)= 0 [if].
tj P*(Xj) 1 t=j

Now suppose h is an arbitrary element of S( %F ) . Define
N
s = 2 h(X)s .

i=1 i i

1

Then g s 3( PI), and
N
x. - h x = 2 h x x. - h x
g< J) < j> 1: ( 1’81( 3) < j>

l

h(x ) - h(xj)

3

Through the use of the interpolation properties of elements of S(-%* )

we can show

(3 - h)‘3)<yi> g‘j’<y1) - h(3)<y1)

= O ’ j=0,ooo,m _1; i=1,ooo,k.

i

k
Thus g - h has N + 2 m1 = n + l zeros, and g E h. Hence, any
i=1

element of S(-%* ) can be written as a linear combination of

{g,...,gN}. These functions are linearly independent, since

38

l
gt(xj) - th. So we have a basis for S( p* ) consisting of

N = n -m+ 1 elements.'

Let {g1,...,gN} be a basis for S( i, ), x = (gl(x),...,gN(x)),

and H{o(x)x I x s X( $30} denote the convex hull of the vectors

N

o(x)§ in E for x e X( $3,) where

a(x) = sgn (f(x) ~5;%;‘—)-).

Then we can prove

 

Theorem 2.2: Let f e Cs[0,oo) and $9, a Kn(f) with 3p* = n. Then

%1 is a best approximation to f if and only if

OsH{o(x)§ I xeX(-11)-*)} .

Proof: Suppose $3, is not abest approximation to f from Kn(f).

Then there exists %5 Kn(f) such that

l l
llf ‘ 'p‘ll ‘ ||f ' 3*" °

If x e X( 1%.") we have

 

a(x) (f(x) - l ) E. If(x) -'1-)-(1;{-)-I

p (x)
1
llf ‘ all

1
llf " 3*“

= If(x) - -—--p*]('x)I

IA

A

l
C(X) (f(X) - m)

39

Thus

 

 

1_ _ 1 = ptix) - p(x)
0 ‘< a(x) (p(x) p*(x) ) a(x) ( p(x)p*(x) )

Since p(x) > 0 we must have

C(X)(1_P%%%) > 0,

and so if we let h = l - 2- we have h e S( %3 ) and o(x)h(x) > 0

pic

1
for all x e X( P} ) .

Now suppose 0 e H{o(x)§ I x s X( %} )}; that is,

r
0 = X A a(x

1 i)fi with 1 such that 21 = 1, A 3_o for all 1,
1=1

i i i i

A
and a(xi)xi e H. Thus

«1

But a(xi)h(xi) > 0 for each i, and not all 11 are zero so we have

r
410(xi)sl(xi). .... : 410(x1)gN(xi)) 4.

l 1

"MP1

1

O
A
“NH

A a(x )h(x )
i 1 i i i
N
A a(x ) Z a g (x ) for some a
1 i i k=1 k k i

k’ k = 1,...,N

II
II MP1

1

r N
= 2 z A o a(x )g (x )
1=1 k=1 1 k 1 k 1

N r
= z z A a a(x )g (x )
k=l 1=1 1 k 1 k 1

r
a Z A a(x )g (x )
k 1 k 1=1 1 1 k 1

I
urazz

 

40
r
which must be zero since 2 l a(x )g (x.) = O for all k. Hence we
i=1 i i k 1

have reached a contradiction, and thus it follows that
(2.3) 0 ¢ H{o(x)§ I x e X( %* )} .

To complete the proof of the theorem, we assume (2.3) holds and

show A; is not best.
P

The set X( fik ) is closed and bounded in [0,w); hence it is
compact. Both 0 and x are continuous on X('%% ); hence a(x)x
is, which implies that {o(x)§ I x s X('%* )} is compact in EN.
Therefore H{o(x)x I x e X( %; )} is compact.

Thus by the theorem on linear inequalities (Cheney [2], p. 19)

there exists c = (cl,...,cN) such that

O < <0(x)x,c> = o(x)c1gl(x) + + o(x)cNgN(x)
= o(x)h(x)
1 2
where h e S( S} ). We write h = p* - 1.
In order to construct a better approximation to f than i}, let
1 + A 1
r = - .

 

 

 

‘ 1 A
p*+)«p (1+A)p*+(l+x)p

We will first show that there exists a >‘o such that p* + 1p > O
on [0,») for any A with IAI $.10 .

Since p* > 0 on [0,m), it has a positive leading coefficient.
Thus there exists A0 such that p* + Ap has a positive leading

coefficient for all A with IAI 5.xo. Then we can find 8 > 0 such

 

41
that the inequalities

p*(X) > 1
p*(x) + Aop(x) > 1

P*(X) - Aop(x) > 1

hold on (B,w). Then as in the proof of theorem 1.2 it is not
difficult to show that p* + Ap > O on (8,”) for any A with
IKI fi-Ao' By choosing lo < l we guarantee that rk is a member
of the set Kn(f).

Further, we can find 5 > 0 and 6 > 0 such that p*(x) 3_e

and Ip(x)I :_6 for all x in [0,8]. 80 if we choose Al so

small that

|Ap(x)| 5. |A|5 < v x s [0,8]

Nlm

for all A with |A| :_A it follows that

1’
o < % < p*<x> +Ap<x>
for all x s [0,8]. Thus if

*
|A| < A0 = min {A0,A1}

we have p* + Ap > 0 on [0,m).

Then

Ap* (‘fii - l )

 

p*(p* + Ap)

 

42

Ah
p* + AP

 

Now let a = min {Ih(x)I

1 Ix

[0,“) ” X

x s X( %} )} , and define

M
II

 

a(x)h(x) > %a and If(x) ‘ p*1(x)I > %IIf - %*II I

N
ll

1 0

Then X1 is open and bounded, and X2 is closed with X(-%* ) C: X1;

1 1
X2 r] X( p* ) - 0. Hence there exists u > 0 such that u < "f - p*"

and f(x) - E;%;y-':'u for all x 5 X2, since f(x) - E;%;7I-+ 0 as

x + m.
* * *
Next we will show that we can choose A1 > 0 with A1 fi-Ao
*
so that for all A with IAI fi-Al we have

1 1 l l
llsvrxll < Illf'3*|| '2'llf"3*|| I-

Let Ko denote the right hand side of this inequality. Then if

*
IAI §_Ao we have p* + Ap 3-5 > 0 and hence

= [Alli-1| < 13—J- 2 -1I

*
Ip* + ApI E p

-1- - r
IP* XI
From the fact that 3p* = n, it follows that

B- - lI < K

p* 1

* *
for some constant K1. Then we choose A1 fi-Ao such that

43‘

2’

11L}; < K
e o
*
whenever IAI :_A1 .
Finally consider any A with IAI :_A:. Then if x e X
we have
f(x)-r(x) < f(x)---!'-— + _1_-rot)
A I — I p*(X)I Ip*(X) A
< u +- -—l*-- r (x)
— IP*(X) A I
1
:— u+ (Hf-341w)
_ -.1_
- llf p*II°
If x 8 X1, we note that
5811 (f(x)-51%?) = sgn (f(x) - r)‘(x)>
since
1 l l
If(x) - p*(X)I 2 ”f p*“ ,
1 l __1_
||p*"rA|| ‘ 2 llf p*||
and
_ .1. .1. _
f — rA - ( f - p ) + ( p* rA ).

Thus if x g x and we take A negative,

1’

44

f(X) - rA(X) C(X) ( f(x} - rA(X) )

= a(x) (f(x) --p;%-}-5-) +o(x) (fig— r)‘(x))

 

 

 

_l logx)h(x)
i If ‘ p*" + p*<x> + ms
1 Ao(x)h(x)
5- “f - 3*“‘+ sup ( p*(X) + Ap(x) )
x€xl
1 1 A
5- "f - 3*" +'§u sup ( p*(X) + Ap(x) )
xsxl
1
< llf ‘3*Il~

Hence for all x s [0,w) and suitable negative values of A we have
1
f " < f " _ o I
ll rAll || p*||
Under the assumptions of theorem 2.2, we have the following lemma,

which will be used in the proof of theorem 3.1.

Lemma 2.4: 0 e H{o(x)x I x e X( fik )} if and only if h(x)o(x) :_O

V x e X(-%* ) and h s S(-%* ) imply h E 0.

Proof: Suppose 0 e H{o(x)x I x s X('%*)} and we have h e S( %% )

with o(x)h(x) :_0 for all x e X( %} ). We will assume h i 0
and derive a contradiction.

Let 21,...,z denote the zeros of h which are contained

t

in X( %k ). Then we must have 21 i yj for any i,j since

45

l 1
f(Yi) - P*(yi) = o and “f - p*" > 0. Observe that t :_n.

1 0
We can find g 5 8(IE* ) with g(zi) = 0(21), i = 1,...,t,

for example we could take

t
8 = 2 0(2 )8
1=1 1 i

where 81(zj) = 611’ g1 e S(-%* ) as in the proof of lemma 2.3.

Then we can show there exists A > 0 such that
a(x} ( h(X) + As(X) ) > 0

for all x e X( %} ).

To do so, let Iz be an open interval containing 21 such that
i

o(x)g(x) > 0 for all x e I with I I] I = ¢, 1 # j.
21 21 zj

Let

Then Y is closed. Since X(-%* ) <2 [0,3] for some 8 > 0, Y
is also bounded and hence compact. Since we have removed the zeros of

h from Y, we have o(x)h(x) > 0 on Y. Let

m = inf {a(x)h(x) I x a Y}

M = inf {o(x)g(x) I x a Y} .

Then m > 0 and M > -m. Thus we can find A > 0 such that m + AM > 0.

46

Then h+Age s<%*) and a(x) (h(x)+Ag(x)) > 0 on mi).
By the theorem on linear inequalities, we cannot have
0 e H{o(x)x x e X('%* )}.
Now assume that h e S( i} ) and o(x)h(x) :_0 for all x e X(-%* )
imply h E 0. If 0 d H{o(x)x x s X( %% )}, then there exists
h e S(-%* ) such that o(x)h(x) > 0 for all x e X('%* ). But then

o(x)h(x) _>_ 0 on X( 1131* ) and so h E 0 which is a contradiction.‘
Section 3: Uniqueness of Best Approximation

Theorem 2.3: If %% is a best approximation to f e C8[0,w) from

 

1
Kn(f), then p* is unique.

Proof: Suppose there exists i-e Kn(f) with

1 l
‘2'" If ‘ 3H “ Hf " 3*"

1
and let x e X( 3%). Then

 

l l
a(X) (f(X) -;;-(-;)-) _>. 0(X) (f(X) -p(x)>

or

 

 

l 1 _ 13*09 - p(x)
0 _<_ C(X) (p(x) " p*(x)) - 0(X) ( p(x)p*(x) )

which implies that a(x) (p*(x) - p(x)) 1 0 on x< %* ).

u Pafi‘

Now suppose (2.1) holds. Then p* - p has m mj zeros

l

47

because of the interpolation conditions. If both (p* - P)(xi)
and (p* - p)(x1+1) are non-zero, then we can produce an additional

zero for p* - p in as in the proof of theorem 2.1. If

(“1’ x1+1)

(p* - p)(xi) ¥ 0. (p* - p)(xi+l)==..- =(p* - p)(x1+r) = 0. (9* - p)(xi+r+l) # 0

for some r > 0 then we have r additional zeros for p* - p in

(xi, xi+r+l)° Thus we have to show that p* - p has at least one more

).

zero in (xi, x1+r+1

To do so, let {yt}teT denote the set of interpolation points

in (x ). First suppose {yt} is empty. Then

i ’ xi+r+l

l+r

sgn a(xi) = (—1) sgn a(x )

i+r+l

and so if r is even, a(xi) and a(x ) have opposite sign and

i+r+1
thus (p* - p)(x1) and (p* - p)(x1+r+1) have opposite sign. This

implies that p* - p has at least one additional zero in ).

(xi’ x1+r+1

If r is odd, the argument is similar.

If {yt} is not empty, we note that

l+r+ 2 m
(2.5) sgn a(xi) = [(-l) teT Isgn o(xi+r+l) ;

and so if r -+ th is even, a(xi) and a(x ) have opposite sign.

i+r+1
Hence (p* - p)(xi) and (p* - p)(x1+r+1) have opposite sign and

again we must have another zero for p* — p in ). The

(xi’ x1+r+1

case r + th odd is similar. Hence we have shown that if (2.1) holds,

then (2.4) implies that p E p*.

 

48

Now if (2.2) holds, then 3p* §_n — l and we have n - m + l
alternation points, so we can produce at least n zeros for p* - p by
arguing as above. Thus if 3p :_n - l we have p* E p. Thus we need only

show that p* - p has an additional zero when 8p = n.

Since
(f(x_m+l)- (x1 ))H* > o
n p n~m+l
* - * * - *
we have (p p)H Z_0 at xh_m+1. When (p p)H > 0 at

xn~m+1 we can show p* - p has at least one additional zero by

arguing as in the proof of theorem 1.2. If (p* — p)H* = 0 at xn-m+l’

and p 1 p*, there must be some r such that

(p* - p)(x ) ¥ 0 .

n-m+l-r

Then using arguments similar to those above along with (2.5) we can show that

* _ .
p p must have an additional zero in (xn-m+l-r’ w).

Thus, in any case, p* E p and hence %* is unique.‘

 

CHAPTER III

CONTINUITY OF THE BEST APPROXIMATION OPERATOR

Section 1: Introduction

We wish to improve the uniqueness theorem for R.n by obtaining
some specific information about how fast "f - %fl increases as 1%
recedes from the best approximation. Thus we will prove a generalization
of the Strong Uniqueness Theorem which is given by Cheney [2] in the
case of generalized rational approximation. The result was extended by
Gilormini [7] in the case of ordinary interpolation, and by Perrie [13]

for osculating interpolation on [a,b].

Section 2: Continuity of the Best Approximation Operator for Rn.

We begin with the Strong Uniqueness Theorem for Rn.

Theorem 3.1: Let f e C:[O,m) “ R.n and suppose '%* is a best ap-

 

proximation to f from R with 3p* = n. Then there exists

n
6 > 0 such that for all %-5 R.n we have
1 l l 1
<3.“ llf pl! 1 fills ‘ pll + If ‘ p*ll °

Proof: We may assume %-#'%*, since if not, take 6 = 1. Define

l 1

"f - - - f - —

M) g ,u u ,.n

 

l 1
"3* ' all

Then 6(‘%-) 3.0 since '%* is best. Let

49

50

: ' _1_ l. *
6 _ inf { 6( p ) p e Rn, p ¥ p }.

If we can show 6 > 0, the theorem will follow.

So assume 6 = 0. Then there exists a sequence

1
(3.2) { Pk }

~ 1
from Rn { P* } such that

5( l- ) + 0 as k + m.
pk

We will show that there exist M1 > 0, M2 < m such that

< 1 <
M1 “"31,“— M2

for all k, in order to apply lemma 1.1. Note that

 

 

l 1
,(1 , n 5, II - Hf" - llf 5"
Pk _- l 1
H5115,"
1 ("HI + ”f - in)
= 1 ' 1 1

15.1 ||5*||+||5kll
“5,"

Hence if I“ i- H I has a subsequence which approaches m,
k

5(‘i )+9'o. Thus there exists M < m such that H l H < M for all
pk 2 pk ‘2

k, and from this it follows that the sequence I "%k - %-H I is
k

bounded.

51

Suppose I "% kHI has a subsequence which approaches zero as

k + m. Denote this subsequence again by I H % H I. Then, since
k

 

 

l
"5' ‘ 5'IH - H 55 H “5 I"
we can find Nl so that
3 l
N5'5IH 5.5H5H Vk: h
or
1 l
:- __________
3 l
'5 H 55 H "5' ‘ 5IH
so that
a If ' in If ‘ 5IH
E.
H 5* H |[5 ' 5'IH
Hence we obtain
2 l 1
5|lf‘5Ill uf-5n (1)
H 55 H "5' ‘ 5'" pk

We will show the left hand side is positive for large k. Since

1
H 5% H2 H 5* H H 5I H— H 5* H "5* 5IH

 

52 I

and I" %- ”Ia»o as k + m, we can find N2 such that for all
k

k 1 N2 we have

2
l l l l l
< —- ‘- < —- —- — —
o 21.5" - ".5" “5* I".
Thus the common denominator on the left hand side in (3.3) is positive for
k_: N . Thus we need only show that there exists some N :_max {N1’ N2}

2

such that for all k :_N we have

2 l l l l l
sllf ‘5Ill |5* '5Ill ' If “5*" "5* II 1 K > °

“—-

where K is some constant. To do so, we will make use of the fact

that in this case we have

If -5.|| : 5M

1
since SI must be a better approximation than the best constant. So

511-511”55-51—115""5*"

:5 ("fin-15k") ("5511-15") 515511"

= 5an 15 n 5115 n "5* u 51" "5,, u +5n5k Ilz‘5l|5* u "f"
> 51f" “55151151," (I 5* u + M)

l 112 "in u 5. ll

 

53

for all k.Z.N if we take N :_max {Nl, N } large enough so that

2

2 l l l l
5" ;In(u 3.1+ M) < 'i‘iIIfllll5*II

whenever k :_N. Thus 6(-l )-+>O, which is a contradiction, and so
k

the assumption I H %‘ H I + 0 is incorrect. Hence there exist
k

constants M1 and M2 such that

1
°<M13II3III1M2<°°

 

for all k. From lemma 1.1 we conclude that there is a subsequence

{ %-} which converges uniformly to '%'e Rn on any closed interval;
k
j
for simplicity denote this subsequence by {'l }.
pk
Suppose x e X( %% ). Then
1 l l l l
6 - -' - -' = f - —' - f "'
( Pk) "pk p*" H pk" H p*“

IV

1 1
O(X)(f(x) -;l:—(£3-) - O(X)(f(x) - m)

= a(x) "5L“' - __l;..
P*(X) Pk(x)

Define c* by the relation

c* {a(x)h(x)} h e s< 5* ), “h" = 1

inf max1
x€x( 3*)

Using continuity and compactness along with lemma 2.4 we can show c* > 0.

54

P
The assumption that 3p* = 11 implies that III-3%- - l" < 00, and hence

Pk

—;'- 1
-Jl-————- = 1
Pk

"5; ‘ 1ll

Thus for any k we can find an )1k 6 X( %* ) such that

 

pk(x)k
P*(xk) _ 1
C(Xk) _>_ c*
"55 - III
or
( )
a(xI) ( E5733; - II .1, c* "S%‘- 1"

1 1
= c*|| - -- ll
pk( p, pk)

1 l
x k

c* "l 1
> ~ — - —
_ *
M2 P pk"
for each k since min ka(x)I _>_-fl . Thus we have

x 2

5( i) (pk(xk)) III-5* - 1:" 3. 552: "5* ' 5k"

or

 

55

l c*
6(-—> :—-—-———
Pk szk(xk)

Since { %'} converges uniformly to %-> O on any closed interval,
k

there exists K > 0 such that for all k,

1
—-—-—— > K
pk(X)
l
for all x e X( 3* ). Thus
*
6( l-) :_ EE-
pk 2

which contradicts the assumption that 6( %-) + 0 as k + «n I
k

The assumption 3p* = n is crucial for the argument above since
if 8p* < n we have elements in S( l} ) of the form 1 - 2* with
P P

3p = n and consequently
ll1 '5" = °°'

We remark that more stringent assumptions than those used in the
Strong Uniqueness Theorem for R;[a,b] given by Cheney [2] are necessary,

since if we consider his theorem in the case where n — 0, we find that

any best approximation §-= %* e Rm satisfies the hypothesis

min {O-ap, m-Bq} = 0,

but the following theorem shows that the inequality (3.1) in the

Strong Uniqueness Theorem will not hold if 3p* :_m — 2.

56

Theorem 3.2: Let I%* be a best approximation to f a c:[0,®) from

R.n with 0 < 3p* :_n - 2. Then the inequality (3.1) from the

Strong Uniqueness Theorem cannot hold for all %-e Rn.

Proof: Assume 0 < 3p* §_n - 2. Let E = “f - %*"° We will construct

a sequence {-l } of elements from R such that I:”f --l “I -+ E as
pk n pk

k+oo but III% -%*III—7l—>0 as k+°°.

and ak > 0 for all k larger than some integer NO. Define

l l

 

 

2
Pk(X) e + [p*(x) - e] I x -2k + ak I
k

Then if we choose 8 > 0 so large that p*(x) > 2e on [8,w) and

p* is increasing on [8,m), we have 'E-%;7 > 0 on [8,m). In
R

addition, we can specify f < %- on [8,w).

We will prove that f %5} +'%% uniformly on [0,8], Which will
k

guarantee that %-(x) > 0 on [0,8] if k is sufficiently large and

 

 

k
hence that -l s R if k > N where N is some integer.
pk n - 1 1
First we observe that
l l = l = .g
- e ...
k _ __________

when k :_No. In addition, if x is fixed, then

 

5 7 1

 

 

lim 1 = l = 1
2
g _ 2 *
k+w pk(x) e + [p*(x) — e] lim x 2k + ak p (x)
k-No 1‘

In fact, to show {'% } + %} uniformly on [0,8], let x 6 [0,8]
k

and consider

 

 

 

 

 

 

 

 

 

1 _ = *(x
pI(x) p*(X)I Ip*(X)I ka(x) I
p*(X) _
< K I 1
— x - k 2
e + [p*(x) - e] 2 + ak
k
*
I P (X) _ 1
= K 2
p*(x> + [p*<x> - e] 52 — 35 + aI
k k
I l
= K — 1
I e x2 2x
1 + (1 - ;;?;3') £5 -‘7: + ak
_ .__1_. = _._Ji__
where K — max Ip*(x)I. If we let M max I1 p*(x) 5
XE[0,B] XE[0.B]
we have, for given a > 0, that
1 — —— —- — 95 + a < M 35E + 335 +
13*(X)Ik k kI — k2 k ak

58

w
w I

if k is sufficiently large. Then if .iL‘<.%

 

 

1 _ 1 <
pk (X) p*(X) I -

 

 

§_
K ( 2K )
<
— = e
.1.
2
l 1
Thus { 3'} +-Sk uniformly on [0,8].

k

1 1 m
Since 3} 6 Rn we must have 'Ek decreasing on [k, ) for large k,

 

 

 

and since
' 2
_ [ p* (x)(.$§;:§El_.+ ak + (p*(x) _ e) (.QQEJEESL ) ]
d l k k
dx pk(x) 2

2
e + [p*(x) - e] £§_:§§l_.+ ak
k

is negative on [k,“) for large k, we have -%- decreasing on [k,m).
k

Thus

 

59

Pklx) -' p*%X)| 5- lple)l + |p*%X)|

__;L__ ._;L__
pkao + p*<k>

[A

E 1
2 + p*<k)

for large k. Similarly, we have

 

1
Pk(x)

 

- fool : lib—I + |f(x)|

A
NIL-:1
+
tons
u
{11

when k is large.
We will now obtain similar estimates on [8,k] for large k.

Let x s [8,k]. Then p*(x) > 2e and ak > 0 so

2
e + [p*(x) - e] { Sfi-ZEEZ—-+ ak} > e
k
and hence

1
pk(X)

 

0 <

1
p*(x)

 

Since 0 < < %- for x > B, it follows that

l _ 1 <
pk(X) p*(X) -

60

on [8,k] for large k; similarly we obtain

sis-f(x) i E
k

on [8.k].

Combining the above facts, we see that

l
{Hf-3M +3 as

and

.1 _.l = max .__l__ _.__l__
"pk p*" ipk(X) P*(X)l
xs[0,w)

 

[V

1 1
lpk(k) - p*(k)|

E 1
> — - _ .
— 2 |p*<k>|

Since {;;%E3}+ O as k + m, there exists an integer N such that

.1 _.l .E
"pk p*“ 3- 4

for all k :.N. 50 if 6 is fixed, there exists k0 such that

l 1 6E
lf‘akll‘llf’3*ll < 7.-
O

and (3.1) cannot hold for all elements of Rn' '

In case 3p* = n, we obtain continuity of the best approximation

 

61

operator in the usual way [2]. We will use Tf to denote the best

approximation to f.

 

Theorem 3.3: Let f e C+[O,m) " R and suppose 2* is a best
0 o n p
approximation to fo from Rn with 3p* = n. Then there exists

A0 such that for all f e C:[O,w) we have
"Tf ' Tfol fi- *0 llf ‘ fol °

Proof: From the Strong Uniqueness Theorem we obtain the existence of a

constant 5 depending only on fo such that

l l l l
llfo " all 3- llfo ' all + 5H; ' all

for any 'l e R . Thus
p n

”f0 - Tf" 1 "f0 — Tfo" + (3|le - no"

or

5HTf ‘ Tfol 5- Hfo ' Tf" ‘ "f0 ‘ TfoH
1H%‘fH+Wf'W‘H%‘T%H
5- Hfo ‘ f" + "Tfo ' f" ' "f0 ‘ Tfo"
5- Hfo ‘ f" + llf ' fol + HTfo ‘ fol ‘ ”f0 ‘ Tfol
= ZW‘fJ°

Hence "Tf - Tfo" _<_-g- "f — f0" .'

 

62

1f f e Rn’ then Tf = f. In this case we can show T is
continuous without making any additional assumptions concerning the
degree of the denominator. The method of proof follows that used

by Werner [17] to establish a similar result for R:[a,b].
Theorem 3.4: Let f e Rn. Then T is continuous at f.

Proof: Suppose { fk} is a sequence from C:[O,m) with {Hfk - f"}+ 0

as k + m. Then

«as.-.

"“1. - fk" 1|le ' fk" = llf ‘ fk“

and thus

”Tfk ' Tf"

[A

"Tfk ‘ fk“ + “Tf ’ fk“

[A

"f ‘ fk" + "f ‘ fk"

2 ”f ' fk"

from whence continuity follows.l

When 3p* 1 n - 1, the question of continuity for T is open.
Attempts to modify the proof of theorem 3.1 when 3p* = n — 1 were not
successful, and construction of a counterexample using techniques

similar to those in the proof of theorem 3.2 apparently was not possible-
Section 3: Results for Kn(f).

Remark: Lemma 1.1 holds if Rn is replaced by Kn(f). The proof

is virtually unchanged except for the verification that % e Kn(f).

63

This is not difficult; it requires using lemma 2.1 along with the
following lemma.

Lemma 3.2: Let {p } be a sequence of polynomials. Then if {p }
__________ k k

converges uniformly to a polynomial p on [a,b], { pk(j) }
converges uniformly to p(j) on [a,b].
The result follows routinely from the fact that uniform convergence
is equivalent to convergence of the respective coefficients.
In addition to lemma 1.1, the proof of the Strong Uniqueness
Theorem for Rn depends in part on the existence of constants M1 and
M2 such that

1
0<M15ll3klliM2<°°

where { i'} was defined by (3.2). The inequality
k

1 l
was essential in the proof.
Obviously, for approximations by functions from Kn(f), the
inequality (1.1) cannot always hold. However, the interpolation

properties allow simplification of this part of the proof and the

inequality (1.1) is not necessary.

Theorem 3.5: Let f e CS[0,w) ~ Kn(f) and suppose fig is the best
approximation to f from Kn(f) with 3p* = n. Then there exists

5 > 0 such that for all %-e Kn(f) we have

1 1 l l
‘3'” llf ' all 1 5H9: ‘ all + Ilf - all -

64

Proof: Assume p # p* and define

l l
f - — f —-—
5(1) = n -lE| u 2*" .
l l
IISVEII

Then 6( %-) :_0 since %} is best. Let

_ 1 1
a _ inf { 5( p ) l p e Kn(f), p # p* }

We will show 6 > 0.

So assume 6 = 0. Then there exists a sequence { 1-} from
pk
~ 1
Kn(f) { 3;} such that
1
5('— ) + 0 as k + m.
Pk

Again we wish to show that there exist constants M1 and M2 such that

<ul "2M <°°

l - pk 2
for all k. The existence of M2 is shown as in the proof of theorem 3.1.
Furthermore, we have
1
—— f(y ) > 0
pk(y1) l
for all k, and hence
1 l
_ = —— > f
“ pk H max IPk(X)| - (yl)
xe{0,m)

So let M1 = f(yl).

65

The remainder of the proof proceeds as in theorem 3.l.|

We previously noted that every f e CS[0,w) does not have a
best approximation from Kn(f). Thus the continuity theorem for the

best approximation operator must include an existence statement.

Theorem 3.6: Let fo 5 és[0,w) ” Kn(fo) and suppose is a best

1
p*
approximation to fo from Kn(fo) fwith 3p* = n. Let Tf denote

the set of best approximations to f e C8[O,w) from Kn(f) and set

= { f 8 &S[O:°°) f(j)(yi) = f§1)(yi), j: 0,...,m -l; i = 1,...,k I .

i

 

Then there exists a neighborhood N of f0 such that Tf is
non-empty for all f e N [l G.
Further, T is continuous in the sense that there exists 8 > 0

such that if f e N l7 G, then

|le ' Tfoll i ‘3 llf " f0" '

Proof: Suppose f e G and consider those %-6 Kn(f) such that

1 1
ll}; ' fll 5- ||E* ‘ fll ‘

By theorem 3.5 there exists 6 > 0 such that

l 1
dfi'aw ila sl'wo“al

l l
i Ilfo " f|| + llf ' 'pll " llfo " all
1 l
5- "fo - f" + "f - 3*” _ ”f0 - 3*“

66

2 ӣ0 ' fII '
II- - — :II< —3‘ If - fll II-i- - fll: II% - fll -
5 l 1
Suppose “f — £0" < E . Then we have I; — 3*" < l for all
%e Kn(f) which satisfy IIf - fin i “f - - pic." We will show that f
has a best approximation from the set
IfieKn<f)I||%"ll<1I’

and hence N can be taken to be

Ifec IIf—foII<%I.

 

E=E( f)=1an"f-l

185}.

Then there exists a sequence { % } C Kn(f) such that
k

     

IIf--kIII +E as k+co,
with "f - — P*>II _ "f — - pk—II > E. Thus for all k
Ilfi k‘II — IIf " p*II + "f"
and
l l
f(Y1) = pk(yl) -<- II 3k II

67

since I %-} C: Kn(f)' Thus by the remark beginning Section 3 we
k

have the existence of a subsequence { %'} and an element %-e Kn(f)
k
:l
such that {'l } +-l uniformly on any closed interval [0,a]. Denote
k
j
the subsequence by { %'} .

R

Then for any x e [0,“), there exists a such that x e [0,a], and

 

“’0 ' pIx)I 5 If“) ’ EjEI + IPkI") _ 51%|

 

1 1 l
i IIf " all + “‘8" lags}? " ml .
xe[0,a]

Since "f --l H -+ E and {'l } +-l uniformly on [0,a] we have
pk pk p

l
p (X)

 

If(x)- I13.

Hence it follows that "f - %II 1 E and i- is a best approximation to f.'

Techniques similar to those above can be used to prove the following

existence theorem in case n is sufficiently large.

Theorem 3.7: Suppose n :_2m. Then each f e CS[O,m) has a best

 

approximation from Kn(f).

Proof: We will produce an element i} e Kn(f) and then restrict the

search for a best approximation to the set

IF

68

l l l
S=I38Kn<f>IIIf-filillf‘a*llI-
The system
= 0 m -l
10) _ <'> ,,
<3>J<yi> - £3071), =1,” 'k

is equivalent to the system

p(j’oi) = (%)(3’(y1),

 

By the theory of Hermite interpolation, there exists a polynomial p0
with 8po §_2m — l which satisfies the latter set of interpolation
constraints. However, po is not necessarily positive on [0,m),
and so we will construct q* = po + sq so that q* > 0 and also
satisfies the constraints.

Let

q(x) = n (x—yi)
=1

P-

Then q has the following properties:
(1) (Kit) :0 on [0,°°)
(11) q(j)(yi) = 0, j = 0,...,mi-l; 1 = 1,...,k
(iii) Bq = 22mi = 2m f_n .

Thus for any 6 > 0, (p0 + eq)(j)(yi) = ( %’)(j)(yi)-

69

Now Sq > 3pc and thus q* = po + eq has a positive leading
coefficient. Hence there exists 8 > 0 so that q*(x) > 1 on
(8,“) with y1,...,yk interior points of [0,8].

In addition, there exists a constant M > 0 such that
Ipo(x)I §_M on [0,6], and since po interpolates positive values
at each yi, there is an interval Ii about yi with po(x) > 0

on Ii for all 1. So if we choose a > 0 so that
min eq(x)] > M
xe[0,B]—UIi

1
then q*(x) > 0 on [o,m), and a} e Kn(f).

To show f has a best approximation from S repeat the

argument used in the proof of theorem 3.6.'

BIBLIOGRAPHY

10.

11.

12.

13.

14.

15.

Arch. Rat. Mech. Anal., Vol. 21 (1966) pp. 391—401.

H

BIBLIOGRAPHY

I. Achieser, Theory of Approximation, Frederick Ungar Publishing
Co., New York, 1964.

W. Cheney, Introduction to Approximation Theory, McGraw—Hill, Inc.,
New York, 1966.

W. Cheney and H. L. Loeb, Generalized Rational Approximation,
Siam J. Numer. Anal. B, Vol. 1 (1964) pp. 11—25.

, 0n the Continuity of Rational Approximation Operators,

Deutsch, On Uniform Approximation with Interpolatory Constraints,
J. Math. Anal. Appl., 24 (1968) pp. 62—79. .

 

Gilormini, Approximation Rationelle avec des Noeuds, C. R.
Acad. Sc. Paris, Serie A, t. 263 (1966) pp. 286-287.

, Continite de l'approximation Rationelle de Tchebycheff

avec des Noeuds, C. R. Acad. Sc. Paris, Serie A, t. 264 (1967)

pp. 359-360.

L. Loeb, Un Centre-example 5 un Resultat de M. Claude Gilormini,
C. R. Acad. Sc. Paris, Serie A, t. 266 (1968) p. 16.

L. Loeb, D. G. Moursund, L. L. Schumaker and G. D. Taylor,
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with Interpolation, Siam J. Numer. Anal., Vol. 6, no. 2 (1969)
pp. 284-293.

Meinardus, A roximation of Functions: Theor and Numerical
Methods, Springer-Verlag, New York. Inc., (1967).

P. Natanson, Constructive Function Theory, Vol. I, Frederick Ungar
Publishing Co., New York, 1964.

. Paszkowski, On Approximation with Nodes, Rozprawy Mat., 14 (1957)

63 pp.

L. Perrie, Uniform Rational Approximation with Osculatory
Interpolation. Doctoral Dissertation, University of Oregon, 1969.

R. Rice, The Approximation of Functions, Vol. I, Addison-Wesley
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J. Rivlin, An Introduction to the Approximation of Functions,
Blaisdell Publishing Co., Waltham, Mass., 1969.

70

71
16. H. E. Salzer, Note on Osculatory Rational Interpolation, Mathematics
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17. H. Werner, 0n the Rational Tschebyscheff Operator, Math.
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