m . ,

m m7 .
‘ N 1

m fimw .

0

M mmm

m Mmpn

. Ravi? »

m mamz

m mwm

H mm ‘

 

 

This is to certify that the

thesis entitled
Theorems on Nonharmonic Analysis

presented by

David Ray Peterson

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mathematics

777 <77/

7/ N’ajoré! pr e

 

Date 8-4- 71

0-7 639

 

 

ABSTRACT

THEOREMS ON NONHARMONIC ANALYSIS

BY

David Ray Peterson

Let A = [X be a complex sequence satisfying

k]
sup [lIm lkl} < a, and inf [ka - xmll > O. The Pdlya
k k,m
kfim

maximum density P(A) of A is defined by,

P(A) = lim lim n(r+t) - n(r) ,
tau |r|4m t

where n(r) is the number of xk with (sgnr) Re Xk E (O,|r|).

Let p and q satisfy, lSpSZSqu and é+é=l.

In chapter I, we show that for r > WP(A), there is

a constant M(A,r) such that

ikkt p 1/p

j .

q 1/q . r
L Zflakl J s M(A,r) [IrIZZake
for finite sequence {ak].

In chapter II, we consider real sequences A satisfying

2P
of symmetric independent random variables on the probability

sipflxk - kl] < k(p) = 2:; , and real sequences [zk(w)]

space (Q,u) such that

sup sup [ka - k + zk(w)|] < 1/2.
k w

David Ray Peterson

i(xk+zk(w))t
We then show that although the sequence {e

may not be free for every w, it is free for almost every
w (i.e. there exists a biorthogonal set). We also show

that if {zk(w)] satisfies

inf {u{w|£k(w) 6 [1-€.1]}} > 0
k

for every 6 > 0, then for almost every w, the sequence

' k k t
{e1( 4-£k(w) (p)) ] has no uniformly bounded biorthogonal

set.

THEOREMS ON NONHARMONIC ANALYSIS

BY

David Ray Peterson

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1971

C» 7 / 7‘? 9"

ACKNOWLEDGMENTS

Professor William Sledd has been of invaluable assistance

in the preparation of this thesis.

ii

TABLE OF CONTENTS

CHAPTER I A NONHARMONIC HAUSDORFF-YOUNG Page
INEQUALITY . . . . . . . . . . . . . . . . 1
Section 1: Introduction . . . . . . . . . . . . 1
Section 2: Theorems on Bounded Biorthogonal
Sets . . . . . . . . . . . . . . . . 3
Section 3: A Hausdorff-Young Inequality . . . . 6
CHAPTER II THE EXISTENCE AND BOUNDEDNESS OF
BIORTHOGONAL SETS . . . . . . . . . . . . 10
Section 1: Introduction . . . . . . . . . . . . 10
Section 2: Biorthogonal Sets . . . . . . . . . 11
Section 3: Bounded Biorthogonal Sets. . . . . . 18
Section 4: Examples . . . . . . . . . . . . . . 24
APPENDIX. . . . . . . . . . . . . . . . . . . . . . . 26
BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . 36

iii

CHAPTER I

A NONHARMONIC HAUSDORFF-YOUNG INEQUALITY

Section 1: Introduction.
In the following, p and q will denote numbers satis-

fying.

+ = l, l < p S 2 S q < a.

bus
QIH

For functions f(t) 6 Lr(-a,a), let

a r l/r
= [ f lf(t)| at]

”f“
Lr(a) -a

Similarly, for a sequence {ak} of complex numbers let

Hfaler = [ z:|ak|r]1/r.

A theorem of Hausdorff and Young [9] states the following:

(i) For f(t) e Lp(-W,w), and

_.II Tr -ikt
(1.1) ak _ 2W [Wf(t)e dt.

Then (2.)1/r Hlaleq s urn p
L

(W)
(ii) If ”(ak}np < a, then there is an f(t) e Lq(-w,w)

satisfying (1.1) and

s (2n)1/q ”(a

(w kHlp.

HfH
Lg

For a complex sequence A = {AR}, the nonharmonic general-

izations of this theorem are conditions (i’) and (ii’).

(i’) There is a constant M(b,A), depending only on
b and A, such that for f(t) E Lp(-b,b), and

-1 b
(1.2) ak = (2b) f f(t)e

“ikkt dt,

then Illa ill 5 M llfll -
(ii’) There is a constant N(b,A) such that if “(ak]Hp‘<°'
then there is an f(t) E Lq(—b,b) for which (1.2) holds and

lit-HM) s quaknlp.

Conditions (i’) and (ii’) have been shown, [1], to
be equivalent to the existence of constants M’(b,A) and
N’(b,A) such that (i’) and (ii”) hold for all finite sequen-
ces [ak}. .

"'> H73 ”kt” M’llf 1H

(1 a e s a .
ikkt
"II I
(n > Illakllq s N Hz: ake up.
Definition 1.1:

For a sequence A, define a Hausdorff-Young radius

Hp(A) by.

HP(A) = inf{b| conditions (i”) and (ii”) are satisfied].

A necessary condition for (i”) is,
(1.3) sup (IIm lkl] < a,
k
and a necessary condition for (ii”) is,
(1.4) Inf {AR—1ml] > o.

kfim
k,m

A will always be assumed to satisfy (1.3) and (1.4).
Since theorem 1.8 implies that (1.3) and (1.4) are sufficient
to imply (i”) over any interval (-b,b), the problem of
generalizing the Hausdorff-Young theorem is reduced to deter—
mining a relationship between Hp(A) and A. This in turn
is equivalent to finding conditions on A under which (ii”)

is valid.

Definition 1.2:
The Polya maximum density of A, denoted by P(A), is

defined by,

 

pm) = 1im lim n(r+t) " “‘9
t... lrhw *—

where n(r) is the number of Xk with (sgnr)Re)\k E (O,|r|).

In 1957, J. P. Kahane [4], showed that for real sequences
A satisfying (1.4), H2(A) = WP(A).

The results of Chapter I demonstrate that for complex
sequences A, l s p s 2, the determination of HP(A) reduces
to the determination of H2(A) for real sequences and there-

fore,

Hpm) = H201) = (rpm.
Section 2: Theorems on Bounded Biorthogonal Sets.
Definition 1.3:

A sequence of functions (gk(t)}, gk 6 Lp(-b,b), has a

biorthogonal set if there exists a set of functions (hj(t)],

hj 6 Lg(-b,b) such that,

b 1 if j = k
(‘ h.(t)g (t)dt = a. =
4b 3 k J'k 0 otherwise.

Definition 1.4:
A sequence {hj(t)] is bounded in Lq(-b,b) if
sup {Hh.H ] < m.
j 3 Lq (b)
Definition 1.5:
Let Rp(A) be the radius defined by,
11kt}

Rp(A) = inf (b|(e has a bounded biorthogonal set in

Lq (—b,b) }.

The connection between biorthogonal sets and conditions

(ii”) is theorem 1.9 which says R1(A) 2 Hp(A).

Notation:
The letter M, with various subscripts, will denote
various constants with dependence on variables denoted in the

usual functional manner, e.g. Mj(e).

Theorem 1.6:

RP(A) = RS(A), l s p s s s a.

Proof:
Since Rp(A) s RS(A) for s s p, it suffices to show
that R1(A) s RéA). For Rm(A) = m there is nothing to

show, so assume RQ(A) < m.

For any 6 > 0, select r1 = Rm(A) + 6, r = r

2

1 + 26,

and (h.(t)}, h. E L1(-r,r), a bounded, biorthogonal set

3
for {elxkt].
r1 izt
Set H.(z) = [' 11.(t)e dt,
3 4r1 3

. 2
Sin 6(z xj)

Fj(z) = Hj(z) Q. (z-xj)6 ) '

 

and f (t) = F F.(x)e_tx dx.
3 4m 3
Since hj E L1(-rl,rl), we have Hj(x) E L¢(-a,m)
rllzl
H.(z) = 0(e ),
3 IzI-w

so that Fj(x) 6 Ll(-m,m) n L2(-a,m) and

(rl+26)|z|
F.(z) =O(e
j lZI-‘Oo:

By a theorem of Paley and Wiener, [8, p. 370],

exist functions kj(t) E L2(-r2,r2) such that

r .
P. (z) = f 2 k. (t)eltz dt,
3 er J

and

there

and because Fj(x) E L1(-m,m), the Plancherel theorem implies

that fj (t) = kj (t) a.e.(t).

The set [fj(t)] satisfies the following properties,

r2 11kt
(a) [r fj(t)e dt = Fj(1k) = 6j'k
2
(b) fj E L (—r2,r2).
(c) sup (Hf.H a ] s sup{HF.H 1 ]
j 3 L (r2) 3' 3 L (-°°.°°)
S suleHjH a } M1.

3 L (-mom)

‘where sup{HH.H ] s sup{Hh.H ] s M . Hence,
j 3 L°(-~.m) j 3 L1(-r1.r1) 2
R1(A) 3 r2 = RQ(A) + 36, for all 6 > O. D

The next theorem, stated without proof by V. D. Golovin
[2] in 1962, says that sequences which are close together

have the same properties in terms of biorthogonal sets.

Theorem 1.7: (V. D. Golovin, [2])
If A and B are sequences satisfying (1.3) and (1.4),
with

Slip ”Ak- Bkl] < m'

then R2(A) = R2(B).

Proof:

See the Appendix.

Section 3: A Hausdorff—Young Inequality.

In the Hausdorff-Young theorem there is a duality be-
tween (i) and (ii), see [9, pp. 101—103], similarily, theorem
1.8 (condition (i”)) will be used in deriving theorem 1.9

(condition (ii”)).

Theorem 1.8:
If A is a sequence satisfying (1.3) and (1.4), then
for any A, there is a constant M(A,A) such that
H2 a eixktll S WA A) Illa 1H
k Lqm ' k p

for finite sequences [ak].

Proof:
Con51der a sequence B = [Bk], where Bk = MBAk and
M3 is large enough to allow each Bk to be written,

- Q + 6 with u [IA I} S M A) d Q - A 1 'f
Bk — k. skp Bk 4( . an - J on Y 1
k

= j,

 

iékt
Expanding e by
iékt = E; (iékt)s '
5:0 5!
then s
' t . Q (ié t)
H23 akelak ll -—- Hz akelfit >3 --s—‘%-— H
Lq(A) s=O ' L9(A)

fl: -0 ‘ ake ék “Lq(A)

m . s .A
5 2' (it) E a eiktB 5
5:0, 3' k k HLH(A)

on s A .Q
s ZDJA%' HZDa 5 5e1 t .
s=O 8' k k HLq(A)

From the Hausdorff-Young theorem,

. A
H23akékselfitqu s (myq (§%-+ 1) ”(akBkslH

(A) p

s (2w)l/q (§%-+ 1) M4(A)S IllaZJHpA

Therefore, 23 a e s (271) q(___ + l) E [a I
H k lquW 2, M ————u (up

= M(A.A) Hlaklup
i6 t 11 t
k H ;x = M3l/qHZDake k H , the theorem
L9(g—) L9(A)
3

Since ”Zlake

is complete.

Theorem 1.9:

R1 (A) 2 Hpm) -

Proof:

Select r1 = R1(A) + 6, r = r1 + 26, and a bounded,

2
biorthogonal set [gk(t)], gk 6 L”(-rl,rl), and let

"A
hk<x) = e l kxx[_5'6](x).

and wk(t) = [mgk(x) hk(t-x)dx = (9k * hk)(t)°

Then wk(t) = O out51de [-r2,r2], and

r2 ixjt a ixjt m ixjt
[r wk(t)e dt = [mhk(t)e dt [mgk(t)e dt = 25 6j,k
2
If [dk] is a finite sequence with “(dklup = 1, then
1'2 q r2 co q
~F IL dk wk(t)| dt = f IE dk f gk(x)hk(t-x)dxl dt
" -r -e0
2 2
r r ix x -ix t
2 2 k k q
s f [I IZIdkgk<x)e e x[t_5'ty§y|ex] dt
—r -r
2 2
r r ix x -i1 t q
k k
s (25)q/p [r2 f 2[Zldkgk(x)e e I dxdt.

2 "r2

Switching the limits of integration and applying theorem 1.8

to the right hand side,

r q r -11 x q
.J‘ 2IZ) dkwk(t)l dt 3 (mug/p M(r2.A)q [ 2 ||[dkgk(x)e “ )Hp dx
_r2 _r2
qr lIm) |
s (26>)q/p M(r2.A)q sup [Hgkllqco ] sup[e l k”)
k r1) k

s (2{>)q/p M5(r2.A)q.

For any finite sequence [ak], the inequality above

yields,
”(akin = sup ()2 akakn
- r ix t
= (25) 1 sup | I 2 ZIdjwj(t) Z)ake k dtl
”{dk3Hp = 1 -r2
‘1 ll ix tn 112: ll 1
s (26) Z} ake sup ( dkwk(t)
p H {dkHlp=1 Lq(r2)
_ i) t
s (26)1/q M5(r2.A) ”Ziake k Hp.
Therefore, H§(A) s R1(A) + 36 for all 6 > O. D

It follows easily from the Hahn-Banach theorem that
Hb(A) 2 Rp(A), so from theorems 1.6 and 1.9, Mp(A) = R2(A).
Theorem 1.7 and Kahane's result for real sequences implies,
H2(A) = wP(A). Noting that theorems 1.8 and 1.9 are valid

for p = 1, theorem 1.10 follows immediately.

Theorem 1.10:

 

For 1 s p s z, Hp(A) = wP(A).

CHAPTER II
THE EXISTENCE AND BOUNDEDNESS
OF BIORTHOGONAL SETS
Section 1: Introduction.
In Chapter I, we showed that for a sequence A and
b'> Hp(A), there is a constant M(b,A) for which,

"u

(2-1) ”la 1“ S M(b.A)H23a e
R q k Lp (b)
for finite sequences [ak}.

N. Levinson [6, theorems XVIII and XIX] has proven the

following theorems.

Theorem 2.1:

If A is a real sequence satisfying,

_ 2:1-
(2-2) Sipllkk RI) 3 D < 2p - k(P)
ixkt _
for some p, 1 < p < 2, then [e } possesses a unique,

bounded, biorthogonal set [hj(t)), hj e Lq(—w,v).

Theorem 2.2:

, There is a sequence A satisfying
(2.3) sup {ka-kl} s k(p).
k
ixkt
for which [e } has no biorthogonal set in Lq(—w,w).

lO

11

Suppose that A satisfies

sup[l1k-k|] < m, inf {lxk-xml] > o,
k k,m
k¢m
so that Hb(A) = W.

. If condition (2.1) were satisfied for b = n, then
[elxkt} would possess a bounded, biorthogonal set. Theorem
2.2 therefore shows that a general theorem regarding a Hausdorff-
Young inequality for b = Hp(A) would have to include a
condition on A similar to (2.2).

In 1963, M. I. Kadec [3], showed that for p = 2, (2.2)
is sufficient to imply (2.1), but the question remains open
for 1 < p < 2.

In this chapter, we consider the existence and bounded—

ness of biorthogonal sets in L9(-w,w), instead of the stronger

condition, (2.1).

Section 2: Biorthogonal Sets

Let L(w) = {Lk(w)] denote a sequence of real, indepen-
dent, symmetric, random variables defined on a probability
space (Q,u).

Two theorems from probability theory, see [7], will be

needed.

Theorem 2.3:
If [fk(w)] is a sequence of independent random variables,

then

:3

n
V f w dw = n f w dw.
firk=1 k” k=1fQ k”

12

Theorem 2,4:

Borel functions of independent random variables are

independent.

Let a.e.(u) mean, "almost everywhere with respect to

the measure u".

Theorem 2.5:

Let L(w) and A be real sequences satisfying,

_ _ 2:1.
(2 4) sip (ka kl} S Dl < k(p) — 2p .
.1
(2.5) sip sup [lxk+ Lk(w) - kl] 5 D2 < 2.
w
i0.k + 2k(w))t
Then, a.e.(u). the sequence (e } has a biortho-

gonal set in Lq(-w,w).

Proof:
@ . .
The sequences {jBk}k=-m , for jBk—j = 1k - j, satisfy
a
(2.4) and (2.5) for the sequences of random variables Lihéw)k=_é

where jzk(w) = Lk+j(w), w1th D and D fixed for all

1 2
j.
Therefore, it suffices to prove that a.e.(u), there is

an f(w,t) e L9(-w,w) for which,

W i(A + 2 (w))t
f f(w,t)e k k dt = 6

o,k'
‘W

Let F(w,z) = lim w 1 — z ,
N" 0 < lkl s N< >‘k + £k(w)>

 

A .
and f(w,t) = lim ] F(w,x)e ltx dx,

Adm -A

whenever the limits exist.

13

For any compact set C in the complex plane, the product
above converges uniformly in 0 x C, so that F(w,z) is
an entire function for every w.

We will show in part (i), that a.e.(u), F(w,x) E Lp(-a,m),
and thus f(w,t) e L9(-e,m) [9, p. 254]. In part (ii), we
show that a.e.(u), f(w,t) has support contained in [-w,v].
Lemma 3.5 in the appendix implies that F(w,Lo(w)) # O, and

therefore, a.e.(u),

ink + Lk(w))t

 

 

_1_ f” f(wjt)e dt _ 5
2W . — o,k'
'TT F(“hko + £0 ((1’))
Part (i): F(w,x) 6 Lp(-o,a) a.e.(u).
Letting F(x) = lim [[1 - x Ip dw,

Tr
N4m0<|k|sN o )‘k+‘k(w)

m P m P
we have, I I |F(w,x)| dxdw = I [|F(w,x)| dwdx
Q-m -W(1

 

 

 

 

 

on p
= lim w 1 - X | dwdx
imJ‘QN-«o O<|k|s NI )‘k+ LkW)
(b 1 ) [m 1' ll x [pd d
y Fatou's emma 3 1m w - w x
-cc N-oco Q O<|k|sN Ak+ J6km")
a X p
(by theorem 2.3) = 1im w [1 - | dwdx
L. N-m O<|k|stQ "k+‘k(‘”)

= I F(x)dx.

To show that F(w,x) E Lp(-a,a), a.e.(u), it is only neces-

sary to show that F(x) 6 L1(—a,m).

14

By lemma 3.3 of the appendix, for the constant K, and

any 9 > 0, e <'%

(2.6) %|1

_ X |p+_1.ll_
xk+zk(w) 2

 

X Ip
in e '

 

5 I1-

 

X Ip
)‘k - Lk (W)
for an appropriate choice of + or -, and Ix-AkI,I1kI 2 Eu

Define a sequence [ek(x)} by,

it: as in lemma 3.3, if Ix-ku,|AkI 2 Eu
€k(x) = I x

which maximizes 1-——-
some tO I k+t

P
I for tE[ D2.D2].
otherwise.

Applying (2.6) and using the symmetry of Lk(w),

 

 

 

 

x 13—}- ____}£__ p _ x p
In” 1k+£k(w)l "2IQUI 1k+1khu>l +I1 1k-zk(w)l 1‘3“”
x p
“ l1-1k+ek(X)|
_ x P
Thus, F(x) s kgo |1 xk + €k(X)I .

For 8 < k(p) D1, the functions Gie(x),

 

 

x
G (x) = W 1 - *——~——
satisfy lemma 3.6. If x satisfies,
(2.7) inf [x-Akztellz6,
k
P
1 x
A + e (X)
k k p l P
then S 3 I1 +‘EI = Ml(6)
x
1 _
Akzl: €

 

 

15

 

 

 

K 1 x p
FLX) _ “2+1 xk+ek(x)
k such that k x
IX-Xklvlkkl 2: 1 ' 717;";

 

 

For both i e, lemma 3.1 implies that the product on
3e:M
the right side is bounded above by M2(IxI + l) 3.

If 6 <'§ , then for every x, (2.7) holds for either

i e so that,

3e:M

(2.8) F(x) s M4(e)(IxI + 1) 2 [IG€(X)Ip + IG_€(X)Ip].

Applying lemma 3.6 to Gi€(x),

2N 3e:M2 - (p-1-2p(Dl+-e))

IN F(x) 5 M5(€)

N
1 - 2p(D1+e)

 

p-1-2p(D1+ e)
6 M

Selecting O < e < and letting hi: 2m, we

 

2
have, for some n > O,

2m+1

IF(X) =ZIm F(x) SMSZZ-mn<m.
-u In 2 m

Part (ii): For ItI > n, f(w,t) = O a.e.(u).

 

 

For complex paths CA having endpoints --An and An'
n
with An 4 m, and z = u + iv,
An —itx
IIf(w,t)Idw = I 1im II F(w,x)e dedw
Q Q -A
n
An -itx
(by Fatou's lemma) 3 lim I II F(w,x)e dedw
-A

16

itz

II
He
5

__ F(W.Z)e- d d
w. w

n

S lim I I IF(w,z)IetVIdzIdw
{2 C
n

= lim I etv I IF(w,z)IdedzI.
C Q

A
n

We only need to show, therefore, that for ItI > N, there

are paths CA for which the latter integral converges to O.
n

Suppose that In - jI S'l, and assume IzI > 1, then

 

 

_ u +<iv
I1k20.jll Ak + ‘k(w)ldw
111 _ u
”'9’ = In klm'l ' knew-u 'ka‘gd'l ww'd‘”

l
2
S U0 kaéEJ'l‘ 1k+zk(w)- -uI ZI %LIk¢E.'1j kakM'J

 

by means of the Cauchy-Schwartz inequality.
Using lemma 3.4 in the same manner as lemma 3.3 was used
in deriving (2.8), and defining the sequence [ek(v)] in

the same manner as before,

 

 

 

. 2
. 1 — 1V 1-
(2 10) £)k#oTj I Ak ‘ u + ‘k(w)l dw S k#2,j| Ak “*‘€k(V”2
Let S(u) = (1 - -—i) 'n (1 - -u-j ), so that
—J kiéopj Ak J + Ck (V)

by lemma 3.5, S(u) 2 M6(§-D2), and

l7

 

1_ u-]+1v

(2.11) [S(u)l 7T ll-A k—i:+§((v)|= u‘j J'kfio j| Ak-j+€k(v)

kaéo. 3'
Defining the sequence [ym] by,

_ . _ . Ii
km 3 + ek(v) for [Am 3| 2 6
Am - j otherwise,

so that lemma 3.6 is satisfied, then

 

2.12 1-11.:1'_i_i_v Isl-iii: SM n, 1_1_1_-_j_+_ix
( ) I -j Ikfio jl 1k“) + ek(v)I 7(€)m#3| Ym I
+e)-17T

(by lemma 3.6) S M8 (e)[IvI+-l](la eWL

Applying lemma 3.6 to (2.8),
l
(w)I 22] S M9<e>1lul+ 114ml" 6’ 1.

 

 

(2.13) I1-_.Ul |
ji‘C L‘J’Q Qk£0 jl _)\k+ %k

Combining (2.9)-(2.13), and selecting e small enough

w I1 u+ggv
+ 1k (00)

- Idw
1'2 kyéO )‘k

+.
s M10<e.n2> (Al-Myldwlill

1.+£.(w Iljf-e-mvl+1>‘”<Iul+1'>'”e”'”'
J 3

u+iv

(IvI4—l)1-n (IuI4—l)l-n eTrlvl

Iu + ivI

s M

 

11(eoD)

For i t > W, the paths CA = [2| 2 = Anele, i6 6 [W.2W]}
n

are sufficient. D

18

Definition 2,6:
A set of vectors E is free in a space S if no
element of E belongs to the closure of the vector subspace

generated by the other elements of E.

Corollary 2.7:
Under the hypotheses of theorem 2.5, the set [e I

is free in Lp(-w,w) a.e.(u).

Section 3: Bounded Biorthogonal Sets.
The existence of biorthogonal sets for the vectors
e I under the rather general conditions of
theorem 2.5 suggests that stronger results concerning closure

properties might exist, being a Riesz basis for example,

but theorem 2.9 states that this is not the case.

Theorem 2.8: [6, p. 5]

If A is a sequence satisfying,

lxml S [m] + k(p).

ixmt
then [e ] is dense in Lp(-w,w).

Theorem 2.9:
Let {zm(w)] be a sequence of real, independent, symmetric,
random variables on the probability space [0,u} such that

for every 6 > O,

19

3(6) = inf {n(wlim(w) e [l-e.1]}} > 0.

m
and
sup sup {ILm(w)I] s 1.
m w
i(m4-Lm(w)k(p))t
Then, a.e. u, the sequence {e I has no

bounded biorthogonal set in Lg(-w,w).

Proof:
im+tmwfldpflt
From theorem 2.5, a.e. u, the sequence [e I
has a biorthogonal set which, by theorem 2.8, is unique. Thus,

we need only show that the biorthogonal functions hj(w,t)

given in theorem 2.5,

 

 

m F.(w,X)e-itxdx
h.(w,t) = . 1
J -a: Fj(w,j+ Lj(w)k(P))
where
F (w.X) = W 1 ‘ _ I - j '
3 W m 3 + 1m(w)k<p) J

are not uniformly bounded in Lq(-w,n).
Fix N > O, and define a sequence of random variables
{um} by.

u = 78 L .(w)sgn j.
“1 C)<IjI:;N m+J
If a sequence [ij of integers is chosen so that

inf {Imj - mnI] > 2(N+l),
J#n
j,n

20

then [um (w)} forms an independent sequence [7, p. 246].
3

Let

m.SN(e:) = (wI -£m+j(w)sgnj E [1 - e, 1] for some j,O<IjI SN],
and

mQN(€) = [wI um(w) < 2N - 6].
Then,

A _§;_
mSN(€) C mSN(€) c mSN( 2N )'
Since u(mSN(e)) = l-u(CmSN(e)) S l-S(e)2N, we have

 

 

n(flm-S (6)) S lim (n(r) S (e))}
j 3 N Mac IszM mj N
1.‘ {m A ( m
3 im u S e
M-Ooo IjISM mj N
(by independence)
= lim [ W “(m §N(€))I
M-ooo IjISM j
=13; I F u( S (“3“”)
Man IjISM mj N 2N
= 0
Taking a sequence 6N 4 O as N a m, “(IJIWNm SN(eN)) O,
N j
so that a.e. u, there is a sequence (mN(w)} for which
w EmN(w)SN(€N)' as N 40°.
The functions HN(w,x) = FmN(w)(w,x+mN(w)) satisfy,
H (w,x) = w 1 - ~5L- ,
N saéo< NBS)

21

where sgn(S) as e [Isl - k(p). Isl — (1 - eN)k(p)]. for
N
O < ISI S N and INBS - SI 5 k(p), otherwise.

From lemma 3.5 of the appendix, HN(w, L (w)k(p)) is

"‘N

bounded away from O and also above, so without loss of
generality, it may be assumed that HN(w, sz(w)k(p)) 4

Mo(w) # 0.

Part (i): l < p < 2.

Select 9 > p. If for the functions H0(x) and h0(t),

 

x
HO(X) = s;0 l — s - sgn(s)k(p) ) '
ho(t) = I” H0(x)e--itx dx,

we have HN(w,x) converging to HO(x) in Lfi(-¢,m),

then,

1
W .mNt a
h (t) — e1 h (t)HN(w. z (w)k(p))
LEVI O mN mN l jg

1

5'
4 O,

as N-an,

a. 9
S MiII IH0(X) _ HN(w'X)l I

and therefore,

W a _3 w a
iw'th(w't“ dt 4 IMO(w)I Ivlh0(t)l dt.

:1
q

However, from [6, pp. 65-67], hO(t) = a cos , for
-1

NIH-

” q
a = I cos?) dt.
-w

22

A
Since for q < q,

A
lehj(w't)lth 2 M2 L lwlhj(w't)le ] '

we have, a.e.“,
A
q/q

w w A
sup lh.(w,t)lth M sup sup [ |h.(t)|q dt]
j In 3 2 j 2 <Q<q IV 3

N

IV

-3 W A q/Q
M sp M(w) h(t)q
22<§ I0 I [leo I]

Part (ii): HN(w,x) converges to Hb(x) in L§(—m,w) for
p < 9 < 2.

For a fixed A > O, |2x| S A - 2, and N 2 A, we have

 

 

 

 

 

<i_.;£_ '
B
in l T N S I
S 2A

1 - x '

C s-sgn(5)k<p))l
s S-Sgn(S)(l-eN)k(p) S+k(p)

ASISI<N I Id Lnll - %1| + Z) I Id ‘n'l ' %“

S-sgn(S)k(P) NS|S|<m S'k(P

 

 

I/\

-A+1 a X a -N+l X
6N [ I” + fA-ld Lnll - 124-1] + £-1+ I“ d gull-“17:"L |

 

 

 

23

Since a O as N 4 m,

6N

(“fig

 

(l- "
S-sgn(S)k(p) '

TT 4 7T
|S|22A IS|22A

uniformly over the interval [-A,A] as N 4 a and thus,
HN(w,x) converges uniformly to H0(x) over [-A,A].

HN(w,x) and HO(x) satisfy lemma 3.6 because,

A
sup { B - S l S k(p) < k(p).
N,S IN S '
so that r [H (x) - ( Q d
' o HN w.X)I x
|t|>A
/\
3 2p I [|H0(x)|§ + [HN(w,x)|Q] dx
|t|>A A
9 .12:21' 2
(by lemma 3.6) s M ( ) A p ( ) .
3 65;

Therefore, HN(w,x) 4 H0(x) in Le(—m,m),

Part (iii): p = 2.
Since HN(w,x) converges to H0(x) uniformly over

[-A,A],

W 2
sup r |h.(w,t)| dt
j-wj

-2 a 2
2 sup IHN(w.zm (w)k(p))l I IHN(va)| dx
N N

-2 A 2
2 ‘M0(w)| sup I |Hb(x)| dx.
A -A

24

For x > O, x # S - %’.

 

S + 1/4 x
In g_§in v/4 sin n(x + 114) = Z) I -d lnll - t'l
w H0(x) F(X + 1/4) S<0 s - 1/4
-3/4
1 .E

S 5J1.” -d lnll - t | dt

_ _ 1. .1

— 2 ln (1+ 3 x).

1 2
Thus, lHo(X)l 2 é.§l§ 17/4431“ W(X4-1A$)' l1 + 4/3"| / .

w n(x + 1/4)

,A 2
so that sup J |H0(x)l = m. D
A -A

Definition 2.19:
A basis [xk} is a Hilbert space H is a Riesz basis
if for sequences {ck}, the series
a:
k.7: Ckxk
=-cn

converge if and only if ”[Ck}H2 < a.

Corollary_g.ll:
For {zm(w)} satisfying the conditions of theorem 2.9,

{ei (m + 1/4 I’m ((1)) t]

the basis is not a Riesz basis in

L2('W,W) a.e.u.

Section 4: Examples
Let Q = % [0,1] have the product probability measure
-00
u: and represent w E Q by w = (---,w_n,°-°,w_l,wo,wl,°'°.

wn'OOI).

25
The Rademacher functions {Rn(w)},

+1, if w E f— 1].
am =1 "
-1, otherwise,

form a sequence of real, independent, symmetric, random

variables.
1 i(k4—Rk(w)z)t
For L <'§, theorem 2.5 says that {e
has, a.e.u, a biorthogonal set, but for k(p) s g 3';

4'
theorem 2.9 says that there is, a.e.u, no bounded biortho-

gonal set.

However, for those w satisfying mm 2‘% for all n,

{ei(k+z)t}

k + Rk(w)z = k + L. and the sequence clearly

has {e1(J-L)t} as a bounded biorthogonal set in L9(-w,w).
On the other hand, for w with w = sgn(n) for all
n.
1 1(k+tk(w)z)t
n, and k(p) s L s-— the sequence {e } has

4!
no biorthogonal set [6, pp. 65-67] in Lq(-w,w).

APPENDIX

APPENDIX

Lemma 3.1 and 3.2 are needed for the proof of theorem

Lemma 3.1:

 

Let y and B be two sequences satisfying,

(i) sip {IIm Yk|.|Im BkI} S S.

(ii) inf {lvk - Ym|.|Bk-Bmll 2 6 > 0.
k,m
k#m

(iii) Sip {lYk - BkI} S T. Y0 = B0 = 0.

Suppose that K(z) = lim W W(z,Bk) = w W(z,Bk) ,
M O< k SM._—_————' k.O ‘—_——_-'
"‘° I ' wank) W(Z.Yk)
.EL
Bk.
6 > 0, there are constants Ml(S,6,e,n,T) = M1’

where W(z,Bk) = l - Then for n > O, and

IV

:ig {'2 _ Ykl]

and M2(S,6) = M not depending on y and B, such that

2'
(3.1) |K(z)| s Mlen'zl,
and
(3.2) |K(x)| S M1(|x| + l)(T+n)M2.
Proof:
For Sk = Bk - Yk

26

 

 

 

 

 

 

7T W(Z,Bk) = “IT I]. + ZSk
|kl2N‘W(2.yk) IZN (z-Yk)6k
2 ex (|z| Z) lskl )
p |k|2N Tz-kaIBQT '
|sk| _2 1/2 _2 1/2

But _ s 'r[ 23 Iz-Y | 1 [ 23 B I ]

|k|2N |z Ykrmkr |k|2N k |k|2N I k

s T M3(S,T,e,6)N-1/2.
Each term W(Z'Bk) is bounded above by (1+wg)(l4w%)==M4(S,T,e,6)
W(zIYk)
so for N1/2 2 T M3,
n
[TM3 + 1]
|K(z)| s M4(S,T,&6) n eanl.
2 6

Suppose that It], Ix], |u|, |u-x| 2 Lv + 2, |t| s'i < l,

and W = u + iv, then [1 - -§§E§-| = i(l - 2xu2 ) as
u +V u +V

x x x x
|1 - 5:;-| = ¢(1 - 5:; ) so that, [1 - 3:; - |1 - u+_iv 2
i [ xu _ x J _ lvl

u2+v2 u-t u2+v2

The left hand side is positive if

 

 

ixt v
2 2 Iv] +|u-t| '
and since '1 - ufi-ivl 2 l1 ’ E_f_;21o we conclude that there
3'
are t1, t2 with Itjl 5-? for which
_§_. -.§ _ X
(3'3) '1 - u+t1' S '1 'WI 5 '1 u+t2 '

For a fixed x, [x] 2 LS2 + 2, if for some k,

(3.4) min{|x-Re~h<L|x~Ref%(L[Re yk|,|Re Bkl] 2 LS2 + 2

28

then there are tk'ek with Itkl'lgkl 5'? for which by
(3-3).

 

 

 

 

(3.5) W(x,Bk) s W(x,Re Bk + tk)
W(x,yk) W(x,Re Yk + 9k)
Let H(x) = W' W(Xafik)
k -——-——_
W(X.Yk)
W(X.Bk)
where ’ denotes that the term. W7;___T is deleted
on

if (3.4) is not satisfied.
2
There are at most 16(LS + 6 + 2)(S + 5 + 1)

 

 

=M ($01406)
W52 5

deleted terms, each of which is bounded above by M4, so
that

Ms 2
(3.6) |K(x)|s (M4) |H(x)|, for |x| 2 LS + 2.

If M6(n) is the number of Yk with Re Yk 6 [n,n+l],
then
4 l
(3.7) M (n) S L+6)(S+6+1) :M (3,6).
6 W62 7

For |x—A|,|A| 2 22 > o,

A+L X B+1 X
(3.8) IA |d znll - {II S 1 IB Id trill - {II

for some integer B satisfying either lB—AI s l or

|A-+t-B s 1.

Selecting L 2 max {3T, g, 5], we have from (3.5),

Re Bk4-tk
ln|H(x)| s 23' f d lnll - fl

Re yk+ l‘ék

29
Re Bk + tk

= Z) 23’ .r d lnll - $1 l
n k for which Re 4, Q
Re yke [n,n+l] Yk k

(by (3.7) and (3.8)

s 2 M7(S,6)(T + 2n) f Id lnll - %1|
|t|,|t-x| 2 l

s M8(S,6)(T + 2n) In (1 + |x|).

Therefore, from (3.6),

M (S.L.6) (T + 2n)M8(S.6)

|K(X)| S M4(T.e.6) 5 (l + |x|) ,

with L depending on n and T.

Lemma 3.2:
Let y and B be sequences as in lemma 3.1. Suppose
that H(z) is an entire function of exponential type a,

H(x) E L2(-m,m), and H(yj) = 5 Then for any “0 > O,

O,j'

there is a constant M9(6, no, a, T, S) = M and an entire

9!
A
function H(z) of exponential type a + no, such that

g(Bj) = 60' , and

j
M) H/fiII 2 s Mgnau 2 .
L (-a,on) L (‘09,00)

Proof:
. . . no lg
U31ng the notation in lemma 3.1, take n < j;-, e <

and select an integer N and a number n1 such that

n
(3.10) an S 7;; and

(3.11) N 2 (T + n)M2 + 2.

30

For KO(z) defined as in lemma 3.1, let
A sin
H(z) =<——T-11—N H(z)KO(z).
By a Paley-Wiener theorem [8, p. 370], there is an
h(t) 6 L2(—a,a) satisfying,

a .
H(z) = I h(t)elzt dt,
-a

 

 

 

 

 

 

so that,
[m] a . mh izt
H (z) =j (1t) (t)e dt,
-a
and
L (-”I”)
Thus, for [Z - Ykl S e,
[m]
|H(Z)I = | a H (YR) (z - )ml
m' Yk
=1
(3.12) s 2|z-ykl eaSa(ea€-1)HHH 2
L (-°I°)
e
10(a e 6)|z - Yk' “H” (_m m)
‘ W(Z.Y-) ‘
Let Kj(z) = KO(Z)N7;:E§) , so that Kj(z)
satisfies lemma 4.1. Noting that for [2 - le s e, we have
.2
W(z,B.) S (l+-6)(1+-6) = M11(T,6)
W(z,yj) le ' z] IYj ' Z]
and by (3.12) we also have,
‘ sin nlz N
19(2): s ManollHllL2(_m . I K342): ————-nlz

31

Since for any 2, |H(z)| s M ”H” ealzl' we conclude
12 L2(-a,a)
that for any 2, there is a j for which,
A a 2 sin n z N ‘
|H(z)| s e I I Ml3(6,no,a,T,S)HHH 2 I‘_fi—21"I | Kj(z)|,
(-°o°) 1

‘when KO(Z) = K0(z).
Applying the uniform bounds of (3.1) and (3.2) and noting

the selection of ml and N in (3.10) and (3.11),

|A< )l H n (“le'z'
H z S M M H e
1 13 L2(_w'¢)
a4—n IZI
o
S M M ”H“ e .
1 13 L2(—..,..)
A (T+n)M2—(T+n)M2 - 2
and |H(x)| s Ml4(nO)M1M13HHH 2 (IXI-tl) .
L (-m'co)
th t H911! M (5 T S) “H”
30 a S on Ia! I 0
2
(-”Im) 15 0 L2 (-m'a)

Proof of Theorem 1.7:
Let y and B be sequences satisfying the conditions
of theorem 1.2. Select r2 > r1 > R2(y), and take a set
2 .
[fk(t)}, fk e L (-rl,rl) Wlth
iy t

r .
s:p{kaH 2 ] < a, and I lfk(t)e 3 dt = 6

_ _ k.j°
L ( r1,rl) r1

r ' t
Let Fk(x) = f 1 fk(t)elz and Hk(z) = Fk(z + yk),

-r
. Q m
and define the sequénces IkujIj=-e' [kvj]j____co by

kuj = Yk+j ' Yk ' ij = Bk+j ’ E’j'
The entire functions Hk(z) satisfy the conditions of

lemma 3.2 for Ikuj} and [kBj] uniformly in k for fixed

32

constants 6,28,2T, a = r1, no = r2 - r1, so therefore,

there exist entire functions fik(z) such that,

r |z|
fik(kvj) = 60,j' |fi<z)| s o<e 2 ).
and
H H S M9 <6 28 2T r :n )H H
gk L2(_a ) 1 o Hk L2(_a a)
Taking Qk(t) E L2(-r2,r2) such that
91km) = :hkme 1"“
then
A r2 iIBk+j_Bk)t
HkaVj) -rz gk(tIe = 60.j
—int iB.t
The set [hk(t)e } is therefore biorthogonal to {e 3 ),
and
r S
II S .2 an
lei L2-—2<r2.r) M9 I‘LZ-lm r1)
Hence, R2(B) S r2 for all r2 > R2(y).

Lemma 3.3:

There is a constant K such that for l S p s 2,

and |t|,|t-x| 2:? + 2, then

.1 _ _s_. _. _ _§_. _.;§_
(3.13) 2|1 t_£| + |1 t+£ s [1 tie ,
for a fixed + or - for [LI 5 1.
Proof:
For It], |t-xl 2 2, [6| 5 l, we have
.;5_ P _ _____ .sL _ §.P
I1"t¢z _ I1 t-6I sI1 SI Is=s (bit)

for some 8’ 6 [t¢z, t-6].

33

Thus, (3.13) is satisfied for 6 = e, if

d P d p s’
(3.14) 5[5§)1-§1 Is: 4-I .I s tfggll —-§I I 1

I fl

2
= 2[_d__|1._.}_<.|pl J(S"'S”),S’”E[S’,S”],
as2 S 5:5”

Let 62 = inf{|6||6 = e satisfies 3.13 for 1 fixed]. Since
the expression in parenthesis on the left side of (3.14)
has constant sign for all 2, there is a choice of sign

for which 18 will satisfy (3.13) for |z| s 1 if

 

O
a > sup {6 ]. From (3.14),
O l
L
2 P
d2}1_§ (SI _Sll)
5! S sup Ill fig ‘ p S=S
s’.S”.sm e [t-z.t+z] .9. 1 _ s. +
as S S=S’ |s=s”

 

 

 

 

 

s sup
s,s’ e [t-L, Izt+1]

S- I P+1, _
(é-iv (27> we... 28>!

 

S l6£2 sup [‘(S’-X)S '+ S, _ 2X|J
s,s’ e [t-1.t+z]

 

2 4(p+l 2
s 162 . + ————-
81;? e [t-L,t+L] mlnI 5"XIvTSII IS’-XI1

This proves lemma 3.3 for K = 200.

Lemma 3.4:

For |t| 2'? + 2, [1| 5 1,

_ 1V '2
t-sgn(t)e

34

Proof:

This is a straightforward calculation.

Lemma 3.5:

If A is a sequence satisfying,

 

 

 

   

 

supH)‘k - kl S Dl <%,
k
and
FA(6)= w (l-f-i.
kfio k
then
-1 w l + [6| - D
IF<6>| .IF<6)Is— 4.).
Proof:
[W6 F -2k
lm sin W5) #3; 1n 1 _ 5 "
k+ Dl
s >3 r ldlnll- l-I
kfio k- -D

 

 

l-D1 m
3 1n <i+l5L1--D16 > .

Lemma 3.6: [6, pp. 48-67]
If A is a sequence and D is a constant such that

sip {ka-kll S D < k(p).

and if F(x) = W (1 - 31?.
kfio Xk

35

then there is an absolute constant M such that

 

I“ |F(x)|pdx S

M
(p-l-ZpD)2 '

 

m - (p-1-2pD)
P M c
'rc IF (X) l dX S (p_1_2pD) I

M N-(p-l-ZpD)
(l-ZpD)

 

2N p
and I [F(x)] 5
N

BIBLIOGRAPHY

BIBLIOGRAPHY

R. P. Boas, Jr., A General Moment Problem, Amer. J.
Math., 63 (1941) pp. 361-370.

V. D. Golovin, On a Riesz Basis of Exponential Functions,
Soviet Math. Dokl. (1962) pp. 920-924.

M. I. Kadec, The Exact Value of the Paley-Wiener Constant,
Dokl. Adad. Nauk SSSR 155 (1964), pp. 1253-1254.

J. P. Kahane, Sur les Fonctions Moyenne-Periodiques
Bornebs, Ann. Inst. Fourier, Grenoble, 7 (1957), pp.
293-314.

, Series de Fourier Aleatoires, Les Presses
de l’Universite’ de Montreal, Montreal, Canada, 1967.

N. Levinson, Gap and Density Theorems, Amer. Math. Soc.
Colloq. Pub1., Vol. 26, Amer. Math. Soc., Providence,
ROIO' 1940.

M. LoeVe, Probability Theory, D. van Nostrand Company,
Inc., Princeton, New Jersey, 1955.

W. Rudin, Real and Complex Analysis, McGraw—Hill, Inc.,
New York, 1966.

A. Zygmund, Trigonometric §§ries, Vol. II, Cambridge
University Press, Cambridge, 1959.

36

   

”'cllfl'l‘flflliljslflflfliifll’flfi'flliu'fljhijflyfififlifilifl