lacunary subsets of orthonormal sets

Analysis Mathematica, 11 (1985), 283--301

Lacunary subsets of orthonormal sets

I. AGAEV

1. Introduction

We consider lacunary subsets of orthonormal sets. Recall certain definitions. 1. An orthonormal set (ONS) of functions ~ = {~pk(X)}k=l, XC[0, 1], is called

lacunary of order p (p >2) or Sp-set if for each polynomial P(x) with respect to this set the following estimate holds:

(1) II~~ - lfP(x)IIL~ -<- c~lle(x)h.

2. An ONS ~ = {q~a(X)}k=l, XC[0, 1], is called a convergence set if each series

at ~Pk(x) with ~ a~ -< k = l k = l

converges almost everywhere. 3. An ONS $ = {CPk(X)}~=l, XC[0, 1], is called a Sidon set if the convergence

of the series ~ [ak] is necessary and sufficient for the embedding k = l

f(x) = Z a~ek(x)EL~. k = l

In the theory of orthogonal series there are well-known results concerning possibilities of extraction, from a given ONS, of subsets with certain additional "advanced" properties. In particular, the following statements are valid.

T h e o r e m A. (BANAen, cf. [3, p. 287] or [1]). I f an ONS {q~n(x)}~=l, xE[0, 1], satisfies the condition

]icp,,I]p <= M, n = l , 2 , . . . , 2 < p < ~ ,

then there exists a subset {Cp,k}k=l which is lacunary of order p.

T h e o r e m B. (MARCINKTEW~CZ, MENCHOV, cf. [5], [6] or [3, p. 357]). Each ONS {Cpk(X)}k=l, XC[0, 1], contains a convergence subset.

Received June 12, 1984.

1 Analysis Matematica

284 I. Agaev

T h e o r e m C. (GAPOSHKIN [2], cf. [10], [11]). I f an ONS {~0,(x)}~~ xE[0, 1], is such that

I~,(x)I <= M, xC[0,1], n = l , 2,. . . ,

then there exists a subset {(,0,k(X)}k~ 1 which is Sidon.

However, the proofs of the above results as well as those of certain other statements of analogous nature leave the problem of the density of the extracted subsets open. Being of independent interest, this problem finds applications in geometry of normed spaces, too.

Below, we consider the following two mutually connected problems. 1. Finite dimensional case. Consider an ONS �9 = n {~p,(x)},= 1, xC[O, 1], II~.llp <-

<=M, n = l , 2, ..., N. Find a lower estimate of the number N' such that a subset {qb,,}~n'l can be extracted from the set ~, satisfying the following requirement:

N'

for each polynomial P(x)= ~ a~9,,(x) the estimate i= l

]lPIIp <= cp, M I1PI[

is valid (c,,a, ~ .... denotes here and in the sequel positive factors depending only on a, fl, 7, ..., and c denotes absolute positive constants whose values may vary).

2. Consider an infinite ONS #={q~,(x)},7=l, II~0nllp-<_m, n = l , 2 , .... What can be said about the density of Sp-subsets extracted from the set #? More exactly, what lower estimates hold for the quantity

~ ( N ) = X 1, N ~ . i: ni<=N

We consider general ONS here. Problems, analogous to 2) for the trigonometric system and other particular sets were treated in [2], [8].

Kasan~ [4, p. 238] obtained the following estimate, exact in order.

T h e o r e m D. Let {9,(x)}n,=~ be an ONS on [0, 1], Iq~,(x)l<=M, xC[0, 1], n = l , 2 , . . . , N . Then there exists a subset {~0,)i~ 1 with N'>=clogN such that

N" for each polynomial P(x) = Z ai q~,~ (x) the following estimate holds:

i= l

PlI= Z ta l <= 4MitPtI=. i=1

The above estimate was obtained by Kashin using a probabilistic approach, more exactly, by taking averages over subsets.

Here we obtain estimates solving, to a certain extent, problems 1) and 2). The main statements (in particular, Theorem 1) are obtained on the base of the method of the proof of Theorem D. We used the outline of the proof of Theorem A, too.

Lacunary subsets of orthonormal sets 285

It is interesting to note that in the case of trigonometric system and even values of p the following exact estimate of ~ was obtained in [8]:

(N) = 0 (N2/P), N -,. ~.

For arbitrary p >2 the above estimate is also valid, but the problem of its exactness remains open.

The case of even values ofp is a special one in the present paper, as well. It may be seen from the contents of part 2 below that in the case of arbitrary p the estimates turn considerably worse.

In the concluding part of the paper we present an example (Theorem 5) due to Gaposhkin, showing that the estimate in Theorem 1 is sharp in the corresponding class of ONS. (We present that example here with a complete proof, using Gaposh- kin's kind permission.) It shows that under the restriction U%llp<=M, n = l , 2, ...,

N it is impossible to guarantee the existence of Sp-subsets of the set {%},=1 with prescribed positive density.

In the proof of our results we use the following version of F. Riesz' theorem (cf. [12, p. 1541).

Theo rem E. Let {%}~~ 1 be an ONS on [0, 1], r=>2, r ' = r / ( r - 1), II~0.11,<=M, n = l , 2 , . . . , r l is some number, 2>=rl>=r ". Then, for fCL,1,

]b,l "('0 -<_ c, mltf]l~}'l) k = l

with

~(rl ) 2 - / 1 = r 1 - r ' ra' bk=o f f ( t ) q~k(t)dt' k = 1 ,2 , . . . .

The proof practically coincides with that of F. Riesz' theorem (cf. [12, pp. 154--155]).

The author expresses his gratitude to B. S. Kashin who supervised the work and to V. F. Gaposhkin for discussions and useful advise.

2. Results

Theo rem 1. Let p > 2 be an even integer, 0 < 5 ~ p - 2 , +={%(x)},N=l an ONS, xE[0, 1], with

II~OnI]~ <= M, n = 1,2, . . . ,N,

where f i=p+6, 0-<M<oo. Then a subset ~ ' = {~P'k}k=lN' set ~, with

N ' = > N :<a), ~(6)--

can be extracted from the

26 p ( p - 2 + 6 ) '

1"

286 I. Agaev

N t

such that for each polynomial P(x)= ~ ak%~(X) the following estimate holds: k = l

N I N r

( Z , ,~v/~ -~ [[PII -<: c,,M,o (Z a~) 1/~. ~ k l ~ - p ~--- k = l k = l

Theorem 2. Let p >2 be an even number, 0<6<=p-2, {~p,(x)}~= 1 an ONS, xC[0, 1], with

[ I ~ o . [ l ~ M , n = 1,2, ...,

where i f : p + & 0 < M < ~ . Furthermore, let ~ be an arbitrary positive number. Then there exists a sequence of integers {rk}k=l possessing the following properties:

N

a) For each polynomial P(x)= ~ ak%k(X ) the estimate k = J

holds true," b)

and consequently,

N N"

( +I)~, n O < r n ~ n n = 1, 2 , . . . ,

c) ~ 1 >= cpN 1/~, N =1, 2, ..., k: rk<~N

where q p- -2+6

f l = l + ( p + ~ ) , q = q ( 6 ) - - 6

Theorem 3. Let p >2 be an arbitrary number and { ( P n ( X ) } ~ = l

xC[0, 1], with

[[~o.]l~<_-m, i f = p + 6 , 6 > 0 , n = l , Z , . . . , N .

an ONS,

~9 N" Then there exists a subset { ,)/=1 with N'>=c log N such that for each polynomial N'

P(x)-- ~ ai%,(x) the estimate i = 1

IIPh ~ l lel[ . <- cp,~,~llPlI~ holds true.

Theorem 4. Let p > 2 be an arbitrary number and {%(x)}~~ an ONS, xC[0, 1], with

Hq~,il~ <_- m, p = p + 6 , 6 > 0 , n = 1 , 2 , . . . .

Furthermore, let ~ be an arbitrary positive number. Then there exists a sequence of integers {rk}~'=l such that

Lacunary subsets of orthonormal sets

N

a) for each polynomial P(x)= 2" ak ep~k(x ) the following estimate is valid k = l

b)

and consequently,

c)

IIPI] IIP]I .=< c., ,ollPH 25"'+" < r, <= 2 ~("+1)1+', n = 1, 2, ...;

1 ~ clogl/O+ON, N = 1, 2, .... k: rk~=N

287

3. Proofs of Theorems 1- -4

Introduce some notations which we use in the proof of Theorem 1. Given a set of numbers G, we call Samples all ordered (increasing) collections

of numbers belonging to a. In particular, if a=[1 , 2, ..., N] and Q = {ki}~"=l is a sample from ~, then 1 <=kl<k2<... <km<=N. We denote by E~ the set of all m-element samples from a=[1, 2, ..., N].

Furthermore, 2" denotes summation over all elements of E~'; given a set

f2, ~ ' denotes summation over all/-element samples from f2. ( ia . . . iD ~

The number of elements of a finite set A is denoted by IA[. Next we define the sets At, A;, A~' where 1 <=l<=p, p being a positive integer.

A t = the set of collections of integers (el, ..., st) such that e v i l , l<=v<=/ and t

z~ev=p; A; = the set of collections of integers (el, ..., e~) such that e~_->l, 1

l<=v<=l, ~e~=p, 2" 1~1; finally, A;'=Az\A;, i .e . A~'= the setofcollections v = l v : e v = l

1

of integers (el, ...,el) suchthat e~_->2, l<=v<=l, ~e~=p. V = I

It is clear that ]At[<=%, l<=l<-p and for l>p/2 we have A~'=0. For the sake of brevity we denote

1

= ( f dx)'. o

The first step towards the proof of Theorem 1 is the following

L e m m a 1. For a fixed collection (5,, ..., el)EAt the following identity holds:

t2E E~" (il..,it)s g2 (ix...it) E E~r

Proof . I t suffices to evaluate the number of samples f2EEff" containing a given collection (il, ..., it)EEl. Each one of these samples is defined by N ' - I elements which can belong to the (N-/)-element set {1, 2, ..., N}\{i~ . . . . , it}.

288 I. Agaev

Hence the number of samples f2EEff' containing a given (i~ ... it) equals Cg'-~ t, which completes the proof of Lemma 1.

L e m m a 2. There ex i s t s a sample f 2 = ~' ~" {n~}~=~s with

p + 3 - 2 N " = [N 2~Pal + l, q - 6 '

such that

(2) p

i ( a ) = Z Z Z v~:~'< t l , . . t 1 = Cp, M, ~. /=1 (~r. .q) EA l ( h , , . q ) C ~

Proof . We prove that for the chosen value of N' the following inequality holds:

1 "~ I(~?) <= c,,M,~. N / ~.J C s acEg"

N' Since [Eft'[ = C i i , the latter inequality implies the statement of the lemma.

We have 1 1 p ~l.,.gl

,--N, Z I ( a ) = Z Z Z Z Ii~...h = '-:u a~Eg' C~' p..~sg' t=~ (W..OEA~(q...OEa

1 v

- Z Z Z Z ~:.::?. -- C~,' ~=~ (~...,,)~A'~ a~efr' ( ~ , . . 0 ~

By Lemma 1

al...~ | N ' - - I Z Z I,~...,, = c~_, Z , 17~::.,?. ~2 E E g ' (il... i~) E Y2 (it...i l) E EN

Evaluate the sum Z li~...~

(ip..it) E E]v

By definition, there exists a number t, such that et= 1. Hence

for (sl...et)EA~.

l<=t<=l, for the collection ( e ~ . . . e 3 E A [

1

= Z zl ~l d x ) 2 (3) Z , If;:,7 ~, ( f p,(x)...e~,(x) <= f f r . .OEE~v (iv..it)E ~- 0

17 1

k = l ( i l . . . i t_ l i t+l . . . i l )EE ~ - 0

Furthermore, by H61der's inequality with q = (p + 8 - 2)/6

N 1

(4) k=lZ ( / -,l"(~ - , , - ~ " " ,+1 , ax)~ <=

N 1

~= ( k z (0 f ~::(~)...~::~(x)~(x)~;::~(x)...~:r


Since

p - - ~ ' + " " p - 1 ~ " " + p - 1 - 1 ,

[Icp~ / 3 = p + 6 , n = l , 2 ..... ,N,

H61der's inequality implies

1

f I~o~,'(x)...q'f*-'(x)cP%'(x)...~o','(x)l("+~)/('-')dx <_ "1 "t -- 1 *t + 1 ~l

0

. . . . "r,lt"" /~(P+~)/(P-- 1)el "r .~t - l t O " I (P+ 5)](P-- 1)~:t- 1 t 5 II "r,t +lu/~(O" [](P+O)/(g--1)~t+1 . ..[[~Oil[[(P+6)/(P--1)~l<__--Cp, M ,8 .

Thus, we can apply Theorem E (see Introduction) to the right-hand side o f (4) with

p + b r ' - /3 p + 6 - 2 r = / 3 , r l - - P--1 ' /3--1 ' c ~ ( r l ) = q - - 6

We arrive at the following estimate of the sum (4):

Since

As

81...~ / Z 1~,...,, <_- ~,,,~,~ Z (i r . .ip E E~ . . . . ' - * (%..'t - ~'t + t-.q) E E~/

]E/~ -1] - - ( ' v / - - 1 < A/'-I--1 -~N = -, , we have

1 Z l(n) <- C~' n~Eg'

< Ct"M'O ~ Z C~'-_/N '-1 N 1-~*'~ <- Ct"M ~ C~'-/N '- lN'-a" = C~ , ' - ' - " ~=le,...~,)ca, , - - C~v" t=l

1 N 1 - 2/q.

N , p

N~lq �9

C~'_-t' ( N - l)! ( N - N') ! (N')! C~" ( N - N ' ) I ( N ' - I ) ! N !

( N - l ) ! (N')! N ' N ' - I N ' - I + I N 't

( N ' - I ) [ N! N N - 1 "'" N - l + l - N' '

the following inequality holds:

1 ~ , N a C~" z~ I(Q) <= cp,u,~ ~ Nl N l - l N 1-~/q -<- cp, ra, a - -

Q E E ~ r ' l = l

I f we substitute N" in the above estimate for its value chosen

N" = [N ~/pq] -[- ] = [N 26/p(p+o-2)] -~ 1,

then, keeping in mind that along with this

(N')P/N ~/q <= c~,

we complete the proof of the assertion of the lemma.

290 I. Agaev

P r o o f o f T h e o r e m 1. Let O : {nl}~'_=l be the sample for which inequality (2) holds. For sake of simplification introduce the notation

{ g i ( X ) } / N ' I = {(Dnl ( X ) } N ' I "

First we note that the left-hand side estimate in Theorem 1 is obvious since

N t

( Z -~'~'~ < IIPll < llPl[ Uk) ' = 9, = p. k = l

Furthermore, all the terms in the inequalities of that theorem are homogeneous in {ak}f'~ and thus it suffices to consider the case when

N' Z a ~ = 1.

k = l

Since p is assumed to be even, the following expansion is valid:

1 N

S = : [i~= 1 a i g i ( x ) f d x =

1 P

of ~. . . afI g~l (x). . . gi, (x) dx, : Z Z Z cil...haq 1=1 @l,..~l)(Al (il...il)EE~r ,

whereby [cq...i,l<-cp, M for each collection (il ... il). Since

we h_',v e

where

p M Az = At UAt,

S < %,M(S'+S")

p 1

S'= z~ Z . Z I%.. .a~l l f gf1(X)'"gq(x) dx , 1=1 (el...,gz) CArl (il...iz)(E]g , 0

p 1

S"= 2 X !~ laf1...afll f g~.~:x~ ,:.,:x~dx I. l=l(~l'"eI)~A~"(il""il)EEtN, 0 11~" : ' " 1 ~ 1 \ "

Let (el ... e~)EA~', then e~->2, l<=v<=l and hence

(5) la~l...aTI l -<= ~ ,.(a?,l... a~" ~,,, <= (a~ + . . . + a~,) u~ = 1. (il...i~) c E}v, (it..iz) c Ely,

Consequently, keeping in mind that for each collection (el ... e~)E At

1

f o q ( X ) ' " g i z ( X ) s 0

S H _~ we get that =__cp, M.


The sum S' can be estimated using the Cauchy--Bunjakovskii inequality:

P

S'-< cp,~(Z Z Z 1%.. a" ~V/2X t = l (,~...h)E.4'z (i1...OEg~,

1 P s s x(Z Z , Z, (f gi,(x)...g,,(x)dx)2) ':'.

/ = 1 ( e l . . . e z ) E A z (il . . . iz)EE~, 0

In analogy to (5), the following relation holds true:

el gl ~. [ai~...ai, 12 ~ 1. (ir..ip E EIv,

Furthermore, using the definitions of {gi}~'=l and I2, we obtain:

1

(Y. Z ZE, (f I/'= l = l ( e i ' " e z ) (N ' (il '"it)E iv, 0

1 P = ~oil(X)...tpi,(x ) dx = 1/2

* = ( e v . . e ; ) g A ~ ( ' x . . . ' 1 ) ~ 9

Thus S = %,M,~ which completes the proof of Theorem 1. Below we need the following

Lemma 3 (cf. [3, p. 287]). Let p be a number, p>2. Then there exist factors A=A(p), B=B(p), depending only on p such that for all f(t)CL,(O, 1), g(t)C CLp(0, 1),

1 1 1

f [f(t)+g(t)] pdt <= f [f(t)[Pdt+p f [f(t)f-2f(t)g(t) dt+ 0 0 0

+ A ]g(t)f dt + B • lf(t)lP-J]g(t)V dt. 0 J = 2 0

Furthermore, a slight sharpening of the proof of Theorem A (cf. [3," p. 288]) provides the following result:

Lemma 4. Let 6>0, e>O, p>2 , {(p,,(x)},~= 1 be an ONS, xE[0, 1], [](p, lip<=M, n = 1, 2, .... Assume that for each sequence

I1=1

1 k k

= n = l

Then {%}~=1 is an S : s e t .

292 I. Agacv

P roof . Let {ai}~= 1 be an arbitrary sequence of numbers with ~a~<=l , i = 1

and set 1

s.(t) = };a,~,(t), i. = f Is.(t)lpdt, i = 1 0

Due to Lemma 3 we have

1 1 1

n = 1 ,2 , . . . .

1.+1 = f IS.+l( t) l"dt <- f [ S . ( t ) f d t w p l a . + l [ f IS~( t ) f -2S. ( t )~P.+l(Odt[+ 0 0 0

1 [p] l

+ a la.+ll p f ko.+l(t)l p dt+B Z [a.+lV f IS.(t)iP-Jl~~ dr. 0 j = 2 0

According to the conditions, we have

1 1

f I~p ' ( t ) l "d t~%'M' If [Sn(t)IP-2Sn(t)q~n+l(t)dt ~ Cp'M'----'--"L-a n(1 + 0/2 " 0 o

Furthermore, la .+xl~la~+l[ ~ for 2<-j<=p. Since

J i n -t ~'--~ -- 1, 2 <-- j <= p,

P P

we get by H61der's inequality

Thus

1

f IS~(t)f-Jlq).+l(t)[ i d t <= 0

1 1

( f Is.(Ol" dO 'p-''/" ( f l~on+l(t)!Pdt) z~ <= cp,~(t,, +1). 0 0

[ n + l ( 1 2 ~ C l a n + l l +cv,ua~+l)[~+Cv, Ma,,+l+ t,,u,a n(1+0/2 ~ L,( l+fln)+~n,

and hence,

Since

~,,+1 <= ( / / (1+~ , ) ) ( z1+ =~). / = 1 k =1

(n=~ 1 ~ t a . + l ] l < ~ 3 < ~. <-- cp,u,~ [a,,+~[~+ n~l+o/~) = cv,u.~ an = cp,u,~, n ~ l n=l n = l

we have In+t<=cp,M,~, n = 1,2 . . . . ,

n : Cp, M~ n = l

which completes the proof of Lemma 4.


P r o o f o f T h e o r e m 2. Using induction, we construct a sequence (r~ = 1) such that for k = 2 , 3, ... the following conditions are satisfied"

( ~ } ; = l

p + 8 - 2 . 1) Rk < Pk <= Rk+l , where -'/~k = k l+(p+")ql~, q - ~ ,

[ / [k--1 [ k--1 [ < Cp,M,j 2) ,=~ 7,rp,,(t) P-~{i__~=l= 7iq~,,(O}~p,k(t)dt : (k_l)(l+O/~

k--1 uniformly in k-1 2_ 7 : (7~}~=1 with ~ ' 7 i - 1.

i=1 Then by Lemma 4 the sequence {rk}~=~ satisfies all assumptions of Theorem 2.

r N Assume that the sequence { k}k=~ has been already defined. For brevity, we X N X N use below the notation {gk( )}k=l= (~0r,,()}k=l" We find a number rN+l, R1v+l<

<rN+~<=R~+2, such that

1

(6) z~ ( f g~(t)...g,~_~(t)q~,~,+z(t)dtf < Cp, M,~ �9 . = N I + s �9 ~1""~p--1 0

(Here and in what follows the summation is taken over all iv, 1 <=i,<=N, v = 1, 2 . . . . , p--1.)

Then conditions 1) and 2) hold for k = N + I. In fact, sincep is even, we obtain N

for all numbers N (7,}i=,., ZT~ = 1 , i=1

1 N N

/ IZi=I "~'t ~Or' (t) [P--2 ( i:IZ ~i~Orl (t))~riv+ i (') dt I =

1 N

1 : , . . . . . Z , ~,...~,.1 o f gil(t)'"gi'-l(t)@r'~-l(t)dt <=

1 <-( z~ 7~....7 ? )l /s( Z ( fg*l( t ) . . .g ip- l ( t )q%,+lf t )dt)2f /2< Cp, M,a _ _ = N ( I + O / ~ �9 ll '"ip--1 II tP--I \ l l ' " I p -- I 0

Thus we need to find such a number rN+l~(RN+i,R,v+2 ] that ( 6 ) i s valid.

Consider the following mean value:

[ R N § 2] I

2 Z ( f gi~(t) . ' .gi ,-~(t)cp,( t)dt) ~ �9 . . . _ n=[Rn+l]+l*1 ~p 1 0

I = .. [R~ + ~] - [RN + d - 1

294 I. Agaev

Interchange the order of summations in the above expression and apply H/51der's inequality to get

[R~r+~] / 1 I<= ([RN+~]-[Ru+~]-I) -=/q Z ( a 2 {,J gh(t)...g,,_~(t)%(t)dt)~f/q.

ix...ip_ ~ n=[ lv+,]+ 0

We apply now Theorem E (like in the course of the proof of Theorem 1). Keeping in mind that

(N+2) ~+~/~(p+~ 1) ~+q/2(p+~) => ( N + 1) q/~(p+~)

we arrive at the estimate

I <= Cp,M, ~ 1 Np_ 1 1

I I ' " l ~ - i < <

Np+. = Cp, M,a N p + 8 = Cp,M,a NI+. �9

Thus, there exists a number rN+lE(Riv+l, RN+Z) for which inequality (6) is valid. This completes the proof of Theorem 2.

To prove Theorems 3 and 4 we additionally need some well-known results. First we introduce certain notations. Let

n

S " = { X = t x " OR" = 1}, = (i=~lx~ , tlxil, IIxlj, ),/3. o =

We say that a set QER" constitutes a g-net of another set GER" (?>0) if for each

xEG there exists YEO such that []x-ylllg<=g.

L e m m a 5. For each 2>0 there exists a g-net ~ c S n of the unit sphere S ~ ( 2; such that If S] <= 1 + .

Denote by f2 the 1/2-net, f2cS" of S" with 1.('21<=5" (such nets exist due to

Lemma 5).

L e m m a 6. Each vector a={ai}L1ES" can be represented as a= ~)oja j j=J.

with 22j<=2, 2j>=O, aJC fs, . / '=1,2 . . . . . j = l

P r o o f o f T h e o r e m 3. Given a set N {~oi}~=1, let n be the number satisfying 4 n ~ .< 4(n+ 1) (D n 5 N = 5 . To prove the theorem it suffices to construct a subset { r)~=l

such that for each polynomial

n

P,(a, t) = Xa,cp,,(t) i=1

the following relation holds true:

(7) ]lP,,(a, t)lI, <= cp,M.oHP,(a , t)[]=, a = (ai)}'=lEf2,


where f 2 c S " denotes the 1/2-net of S" with [O[N5". In fact, given an arbitrary _ _ n n vector a-(a~)~=lES, it follows from Lemma 6 that

a = ~ 2 j a J, Z2j<-2 , 2 j ~ 0 , aSEI2, j = l , 2,.... y = l j = l

But the latter and (7) imply

[j--~l " <= IlP"(a' t)llp = I ~jPn(a'. t)[[, Z 2j [[P.(ai. t)H v <_-- 2c,.~. n. j = l

Before constructing the required subset {~0,,}~=1 , introduce some useful notations.

Let, as above, f2 stand for the 1/2-net of S", f2cS" , 1f2[<=5 ", and for l~_l<=n put

g2 t = {a' = (a,)~=aER t, a = (a~)~=lEO},

i.e., I2 z is the projection of the set f2 onto the first l coordinates. Clearly, [f2~[<= [g2], l<-l<-_n. Let r l = l . Furthermore, assume that the indices rl . . . . ,rk (k<n) has been already constructed, and rsE((s-1)53"+l, s53") for s<=k. Next we prove that there exists an index

r~+lC(kS"+ 1, (k+l )5 ~")

such that the following estimate is valid:

<: Cp,M,6 ( 8 ) Srg+l ~- Srk+l,k : Hq ,

where 1

~m,k = sup{ f IP~(~, t)j,-~P~(~, t)~Om(t)dtq: a = (a~)~=lEg2k}, 0

k Pk(a, t) = Z ai%~(t).

i=1

In fact, consider the following mean value:

(k -~ 1)53n

S=[ (k+ l )5~" -kSS"- l ] -1 X s~,~ <_-- m=kf3n +1

(k+1)53n 1

< = [ ( k + l ) S s " - k s ~ " - l ] -~ Z Z I f lP~(a, Ol'-~P~(a,O~m(t)dt ~. m : k 5 3 n ~ - i a ~ k

296 I. Agaev

As above, use Theorem E, to get

(k+~)~a. 1/ ] S <= [53"- 11 -~ X Z IPk( a, t ) f -~Pk( a, t)fPm(t) dtq <= aEO~ m=l~53n+l 0

1 _ c~,u,atr5a"- 1] -z " ~ ( f IP~(a, t)l(~-~)(v+~)/(P-~)dt) ~(p-~)/(~+~) <=

We have

and thus

C [~3n Z !I P. t" t~)lq(P-- a E Y~

i i __ ~l~o,,lI,~M, n : l, 2, ..., N; I(~,l-<=5 ", l ~ k ~ n ,

. S _-< cp,~,a[5a"-1]-15"n ~(p-1) =< c~,~t,a -~ .

This implies the existence of the index rk+~E(k5a"+ 1, ( k + 1)5 s") above mentio- ned, for which inequality (8) holds. Now we proceed to show that estimate (7) is valid for the subset {~o,~}~= t obtained.

Due to Lemma 3

1 /1

I n : f li~= ai~lgrt (O[ p dt ~ ' n - x ( 1 + Cp, Mazn) +Ct,,~(a~n + la,,l sl~).

Continuing this process for / , ,_,,/ ,_=, . . . , /1 we get

where

Since

in <= (H(1 +or,))(/1+ fi~), 1 i ~ l

=, = C~,,Ma~, ~, = Cp M(a~+la,ls~gq), 1 <_- i <__-- n.

k = l i = l i=1 "~ ~ CP'M'a '

it follows that 1,~%,u,a, i.e. (7) is valid. This completes the proof of Theorem 3.

P r o o f o f T h e o r e m 4. Let

Mk=~sk*+~ k = l , 2,

It follows from Lemma 4 that in order to prove Theorem 4 it suffices to find a sequence {rk}~,, r , = 1, such that the following relations are valid:

Lacunaly subsets of orthonormal sets 297

1) Mk+l<rk+~<=Mk+z, k = l , 2 , . . . ,

2) for each polynomial k

ek(a, t) = z~ ai%,(t) i=1

we have

k with ~ a~ = 1

i=1

1

(9) f IP~O, t)[p-2Pk( a, t)%k+x(t)dtq < c"M'a'-----------L" : kq 0

Assume that the s e q u e n c e {ri}~= 1 has been already constructed. Next we define the number rN+ 1 satisfying 1 ) a n d 2 ) f o r k = N + l .

By Lemma 5 there exists a 2-N%net f2 on the sphere S N with

[g2[ <= (3.2~=) rr ~ 28Nl+L

First we find an index rN+l, MN+I<rN+I<=MN+2, such that inequality (9) holds for k = N and all a=(ai)~=lCt2.

Consider the following mean value:

1

T = (MN+2-MN+I-1 ) -1 '-~*= ~ f [PN(a, t)[P-2PN(a, t)9,,(t)dt q, n = M ~ + l + l aEO g

where q=(p+6-2)/6. Using Theorem E, we get

M~r+2 1

T=(MN+2-MN+x-1) -1 ~ Z f lPN(a,t)lP-ZPu(a,t)%(t) dtq<= aEI2 n=M~c+x+l

1

<= C,.M.,(MN+~--MN+I-- 0 -1 z ( f IPN( a, t)] ("-1)`"+a)/("-1) dt) q("-l,/O'+') <= aE 0

<= Cp.M.a(MN+2--M~+~--1) -1 2 !IPN(a. t ) l l ~ g 1). as

Since il~0.11~<=M, n= l , 2, . . . ,

M N + 2 - M N + I - 1 = 25(N+l)~+'-25m+~ _-> c25m+~, and

IOl -<- 23m+~, we get the following estimate for T:

T <= 2-sm+*Cp,M, a23u'+'N(P-1)q ~ C~,M,~/N q.

Thus, there exists an index rN+l, MN+~<rN+I<=MN+2, for which (9) is valid with k = N and a={ai}~=lCf2.

Consider an arbitrary vector a = {af}~=~E S N. Then there exists an element a~ f2' such that

a = a ~ a ~ = {a~}~=,, e = {el}~v=a, lia[[tf <= 2-NL

298 I. Agaev

Evaluate the following difference:

1

= I f [PN(a, t)IP-~PN(a, t )%~+l( t )d t - 0

1

- f IPN( a~ tlf-~PN(a ~ t)%..+l(t)dt =

0 1

= If (leN(a, t ) f -1 sign PN(a, t) --IPN(a ~ t ) f -1 sign PN(a ~ t))%~,+l(t ) d t . o

Note that for all a, b and 0 > 1.

(10) [lal ~ sign a -[b] ~ sign b[ <: c(O)[a-b[(la[~ Ib[~

(10) implies 1

a ~ ~.,~ f le~(~, tl[(TeN(a, t ) f -~+leN(a ~ tlf-Z)lq~,,+l(t)[ dt = 0

1

= ~.,~, ( f IP~(~, 011PN( a, t)l,-~[q~r~,+~(t)[ dt+ 0

1

+ f IPN(~, t)t [PN(a ~ t)f-@,~,+~(t)] dt) - (I~+Ie). 0

Since 1/1) + lip + (p - 2)/1) = 1, H61der's inequality implies:

1 1 1

I1 <= ( f IP~(~.t)l, at)*/"(f leN(a.t)l,at) "-~'/" ( f le,~+~(t)l, dt) ~/" 0 0 0

Now, it suffices to apply the following rough estimate (we keep in mind that [1~o,[1~<= <-M, [l~llzf<-2-N~

I~ <-_ cp'gNNP-22 n" <= Cp'M2-~ <= Cp'M'*N

Is is estimated analogously. Thus, 2) holds for each vector a = {a,}~=aCS N and for k = N , which completes

the proof of Theorem 4.

4. An example

T h e o r e m 5 (GAPOSHKIN). Let p>=2 and 6>=0 be fixed. There exists a quantity Mp,6>0 such that f o reach N>:I there is an ONS {cpl, ...,cpn} on [0, 1] possessing the following properties:

1) II~,,II,+~ ~ mp,~, k = 1, 2, ..., N;


2) each Sp(C)-subset of it contains no more than [2cZN ~] functions, where

26 a = c~(5) - p ( p - 2 + 3 )

(an ONS is called an Sp(C)-set i f we can take C~=C in (1)). Note that the value of e(5) in the above assertion coincides with that of Theo-

rem 1. Thus Theorem 5, being applied to even p > 2 and 0<6<_-p-2, demonstra- tes the sharpness of Theorem 1.

We make use of the following assertion (cf. [7], [9]).

T h e o r e m F (ScnuR). I f {q~l, .... 9N} is a set of functions defined on [0, 1/2] such that for all ak, k = 1, 2, ..., N,

1 N N

/ Ik~=l ~ k=lZa~"

then there exists an extension {fk}kU=i to an ONS on the segment [0, 1] and, besides,

(11) A ( x ) = X b~,R,(x), x~ ~ , , i=1

where the bki are certain coefficients with

k

Xb~, ,< l ( k = l , 2 , . . . , N ) i=1

and {Ri(x)} is anarbitrary ONS on [ 1 , 1 ] .

P r o o f o f T h e o r e m 5. Let

Then

~o~(x) =

p+5 7 = p - 2 + 6 '

N~/(,-2+a), xE [0, 21--N-,],

o (Zk=, akc&(X))2dx <= T ak)=N2/(P-2+a)N-~' -<= --2 k=lZak'

and due to Theorem F, (11).

{Cpk}~V=l can be extended to an ONS on [0, 1] by formulae

2 Analysis Matematica

300 I. Agaev

If now we take for R,(x) the Rademacher functions I/-ffri(x), ri(x)= sign sin 2*~cx, then using Khintchine's inequality we obtain:

f Ie, l.+'dx= f Ckii/2ri(x)'+ndxN] o 0 112

< L (p+~)/2--1 1 k 1 +2(~+~)12_~cp+o,

where I p + b ~W+a)/~

{q~k}k=l, where we can take Mp,~= 1/p-t - 6 + 2. Thus property I) is satisfied for the set N Let {~o.~, ~o.~ . . . . , ~o.~} be an arbitrary Sv(C)-subset of that set consisting of

R functions. Then for ak = -- 1 the following inequality holds:

1 R f %(x)t" dx <= d'R pt . 0 =

By construction

Thus

~0,~(x) = R N ~ -2+a)-~, xC , N ~ . k=l

1 -~ N - r R p N@-~+~) -~ <=_ cPRP/~,

R ~= 22/PcZN ~ ~ 2c~N ~

which completes the proof.

C o r o l l a r y . For all p > 2 and N~=I there exists an ONS {qh,-.-, q~lr on [0, 1] such that II~Okl[p~Mp ( l ~ k ~ N ) and each o f its Sp(C)-subsets contains no

more than [2C 2] functions.

References

[11 S. BANAO~, Sur les s6ries lacunaires, Bull. Acad. Polonaise (1933), 149--154. [21 B. q). F a n o x ~ x ~ , J/aryHap~rsie p~J~I ~ ~eaa~c~cn~Ie ~y~IoXnn, Ycnexu sname~, nayK, 21

(6) (1966), 3--82. [3] C. Ka~Max~ ~ F. I / I T e ~ r a y c , Teopun opmoeona~bnbZx pabos, ~3MaTrl~3 (Moc~Ba,

1958). [41 13. C. K a m ~ ~ A. A, C a a x a n , Opmoeouaabubte pflf)bt, HayKa (Mocr~a, 1984). [5] J. MARCL~qKrZVvaCZ, Sur la convergence des s6ries orthogonales, Studia Math., 6 (1936) 39----45.

Lacunary subsets of orthonormal sets 30i

[6] D. MENCI-IOFF, Gut la convergence et la sommation des s6ries de fonctions orthogonales, Bull.

Soc. Math. France, 64 (1936), 147--170. [7] H. RADEMACHER, Einige Siitze tiber Reihen yon allgemeinen Orthogonal-funktionen, Math.

Ann., 87 (1922), 112--138. [8] W. RUDIN, Trigonometric series with gaps, J. o f 21~rath. and Mech., 9 (2) (1960), 203--227. [9] I. SCHUR, Ober endliche Gruppen und Hermitesche Formen, Math. Zeitschr., 1 (1)(1918),

184---207. [10] S. SIDON, Uber Orthogonalsysteme, Comp. Math., 7 (1940), 372--375. [ll] S. SIDON, t)ber orthogonale Entwieklungen, Acta Sci. 3,Iath. (Szeged), 10 (1943), 206--253. [12] A. ZYGMUND, Trigonometric series, University Press (Cambridge, 1959) - - A. 3 } I r i y H ~ , Tpu-

eono.uempuuec~cue pabst, T. 2, Map (Moc~a3a, 1965).

J'iaKyHapSble IIO~CHCTeMbl 0pTOHOpMHp0BaHltblX CHCTeM

H . A F A E B

B pa6oTe paccMaTprlBamTca Sp-IIO/ICldCTeMt, I O.H.C. B ~IaCTI~OCTI, I, jlotcaa~tBaeTc~l caezyloma~ TeopeMa, KoTopaa Heyc~an~ieMa. TeopeMa . Ilycmb p ~ 2 - - uemnoe ~uc.qo, J - npousao~bnoe ~uc~o, 0<J<=p--2, # =

~{~0,(x)}~=, - - O. tL C., x6[0,1], npuuem ]lr n = l , 2 . . . . . N, ebe ff =p + J, 0 < M < ~ o.

u~ cucmemT, l # .~o~cno abzgpamt, noOcucme~ty q~" = ~v, N'~--N "r ct(6)=. ,_ 2o+ ~., ~ Toeba {~,.~}~=~, PLP-- o)

N '

maKyto, ~mo 3:m :lm6oeo no~Tuuo:~ta P(x )= ~' ak fP.k(X) u~,teem mecmo otiemca k = l

3 / ' /g"

tk~ ~ ~ j - I l e l l .

(%, M, ~ - - nocmonnnaa, zaaueau4aa mo~bro om p, M, J, no ne om N u.~u rcoao~O6ut/uenmoe noau-

Ho3la) .

rlpIIBO~tTCll II jlpyFBe pe3yJII~TaTI,I a~aJIormmoro xapaKTepa.

CCCP, MOCKBA 117 234 MOCKOBCKHfI FOCY.ZIAPCTBEHHGIf4 YHI'IBEPCHTET MEXAH]rIKO -MATE MATH q E CKHIYI ~AKYJ/LTET

2*

lacunary subsets of orthonormal sets

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