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Systems of random variables equivalent in distribution to the Rademacher system and -

closed representability of Banach couples

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Sbornik: Mathematics 191:6 779807 c2000 RAS(DoM) and LMS

Matematicheski Sbornik 191:6 330 DOI: 10.1070/SM2000v191n06ABEH000481

Systems of random variables equivalent

in distribution to the Rademacher system

and KKK-closed representability of Banach couples

S. V. Astashkin

Abstract. Necessary and sufficient conditions ensuring that one can select from asystem {fn}

n=1 of random variables on a probability space (,,P) a subsystem{i}

i=1 equivalent in distribution to the Rademacher system on [0, 1] are found. Inparticular, this is always possible if {fn}n=1 is a uniformly bounded orthonormalsequence. The main role in the proof is played by the connection (discovered inthis paper) between the equivalence in distribution of random variables and thebehaviour of the Lp-norms of the corresponding polynomials.

An application of the results obtained to the study of the K-closed representabil-ity of Banach couples is presented.

Bibliography: 26 titles.

Introduction

We say that systems {fn}n=1 and {gn}

n=1 of random variables defined on prob-ability spaces (, ,P) and (, ,P), respectively, are equivalent in distribution

(we write {fn}P {gn}) if there exists C > 0 such that for arbitrary m N, an R

(n = 1, 2, . . . , m), and z > 0 we have

C1P

mn=1

anfn() > C z P

mn=1

angn() > z

CP

mn=1

anfn()

> C1z

.

If only the left-hand side of this relation holds, then we say that the system{fn}

n=1 is majorized in distribution by the system {gn}

n=1. In a similar way onecan introduce these concepts in the case of finite collections {fn}Nn=1 and {gn}

Nn=1.

As shown in [1], from a uniformly bounded and orthonormal (in L2) sequence ofrandom variables {fn}n=1 one can always select a subsequence {fnk}

k=1 majorizedin distribution by the Rademacher system {rn}n=1, where

rn(x) = signsin(2n1

x) (x [0, 1]).

AMS 1991 Mathematics Subject Classification. Primary 28A20, 60E99.


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780 S. V. Astashkin

The main aim of the present paper is to prove that, under the same assumptions,it is possible to select a subsequence that is not merely majorized, but also equiv-alent to the system {rn}n=1. Moreover, we find necessary and sufficient conditionsensuring that a system of random variables contains a subsystem equivalent in dis-tribution to the Rademacher system. In the case of finite systems we are interestedin the density of their subsystems with a similar property.

Problems related to the selection of almost independent subsystems (that is,subsystems with a certain property characteristic for systems of independent func-tions) are discussed in detail in Gaposhkins well-known paper Lacunary seriesand independent functions [2], which we shall repeatedly refer to in what follows.We recall the most important results relating to the integration of the sum of aseries in a fixed system of functions and to its absolute convergence.

In view of a classical inequality established by Khintchine [3], the sum of aseries

n=1 anrn(x) (x [0, 1]) belongs to all Lp-spaces (p > 2), provided that

the sequence of coefficients a = (an)n=1 belongs to l2. For lacunary trigonometricseries a similar result was obtained by Zygmund [4]. In this connection, but slightlylater, Banach and Sidon introduced the concept of a lacunary system of order p,or briefly, Sp-system. A sequence of random variables {fn}n=1, fn Lp (p > 2), iscalled an Sp-system if m

n=1

anfnp

Kp mn=1

anfn2

, (0.1)

where the constant Kp > 0 is independent of m N and an R (n = 1, 2, . . . , m).If{fn} is an Sp-system for each p > 2, then it is called an S-system.

Banach proved that one can select an Sp-subsystem from each orthonormalsystem {fn} of measurable functions on [0, 1] such that lim supn fnp < ([5] or [6], 7.2). This, in particular, means that one can select an S-subsystemfrom each uniformly bounded system of functions. Here is a more precise resultdue to Stechkin [2], Theorem 1.3.1.

Theorem A. Let{fn}n=1 be a system of measurable functions on[0, 1] and assumethat p > 2. Then one can select from {fn}n=1 an Sp-subsystem if and only if thereexists a subsequence {fnk} such that

(1) fnkp D (k = 1, 2, . . .);(2) fnk 0 weakly in L2.

We now proceed to a brief discussion of another property of lacunary series, theirabsolute convergence. Well known for trigonometric series is Sidons theorem onthe absolute convergence of a lacunary (in the sense of Hadamard) Fourier series ofa bounded function. Zygmund established a local version of this result [7]. Similarresults were later obtained for the Rademacher system and certain other speciallacunary systems of functions [8]. The following concepts were introduced as anatural generalization of these results. A sequence of random variables {fn}n=1 iscalled a Sidon system if

m

n=1 |an| Cm

n=1anfn, (0.2)where the constant C > 0 is independent of m N and an R (n = 1, 2, . . . , m).


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Systems of random variables 781

A system {fn}n=1 of measurable functions on [0, 1] is called a SidonZygmundsystem if there exists a subset E of [0, 1] of positive Lebesgue measure |E| such thatfor each interval I, |I E| > 0, the condition n=1 anfn(x) L(I) implies that(an)n=1 l1. If one can set E = [0, 1] in this definition, then we say that {fn}n=1is a SidonZygmund system in the narrow sense.

Broad sufficient conditions ensuring the existence of Sidon and SidonZygmund

subsystems have been obtained by Gaposhkin [2], Theorems 1.4.1 and 1.4.2.

Theorem B. Let{fn}n=1 be a system of measurable functions on [0, 1] such that(1) fn2 = 1 (n = 1, 2, . . .);(2) |fn(x)| D (n = 1, 2, . . .; x [0, 1]);(3) there exists a subsequence {fnk} {fn} such that fnk 0 weakly in L2.

Then one can select from {fn}n=1 a Sidon subsequence.Theorem B. (a) Under the assumptions of Theorem B one can select from{fn}n=1 a SidonZygmund subsystem.

(b) if the subsequence in assumption (3) of Theorem B satisfies additionally thecondition

lim inf

k F f2nk

(x) dx > 0 for each F

[0, 1],

|F

|> 0, (0.3)

then one can select from{fn}n=1 a SidonZygmund subsystem in the narrow sense.It follows from the above definitions and the properties of the Rademacher system

that if{fn} P {rn}, then(1) {fn} is an S-system and the constant Kp in inequality (0.1) has the same

growth order in p, as p , as in the case of the Rademacher system, thatis, Kp p ;

(2) {fn} is a Sidon system.Thus, the results established below in 3 ensure nice properties of the selectedsubsystem in both respects, improving and complementing the above-stated results.

Alongside the problems of the selection of infinite subsequences with proper-ties characteristic for systems of independent functions, problems of another kind,treating finite collections of random variables, are also of interest. Let {fn}Nn=1 bea collection of random variables. We are looking for a possibly large s = s(N) suchthat there exists a collection {fni}si=1 {fn}Nn=1 satisfying a certain condition oflacunarity with constant independent of N. We state here just one result on thedensity of finite Sidon subsystems, which was proved by Kashin (see [9] or [10], 8.4,Theorem 9).

Theorem C. Let {fn}Nn=1 be an orthonormal collection of functions on [0, 1]such that |fn(x)| D (n = 1, . . . , N ; x [0, 1]). Then there exists a collection{fni}si=1 {fn}Nn=1 (1 n1 < < ns N) such that s max{[ 16 log2 N]; 1} and

for allai R (i = 1, . . . , s) the following inequality holds:

D1 si=1

aifni

s

i=1

|ai| 4D si=1

aifni

.


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782 S. V. Astashkin

In the present paper, under the same assumptions, we prove a result on the selec-tion of a subsystem {fni}si=1 of the same density that satisfies a stronger condition:it is equivalent to the system {ri}si=1 with constant dependent only on D.

This paper consists of five sections. In 1 we collect concepts of operator inter-polation theory used in what follows (see [11] for greater detail). For completenesswe also present there the proof of a formula from [12] for the K-functional cor-

responding to the pair (l1, l2). In 2 we prove a criterion for the equivalence indistribution of an arbitrary sequence of random variables and the Rademacher sys-tem and derive consequences for some particular cases. The central results of thispaper, on the selection of subsystems equivalent in distribution to the Rademachersystem, are the subject of 3. The next section is devoted to similar problems forfinite collections of functions. Finally, in 5 we apply these results to the study ofthe K-closed representability of Banach couples.

In conclusion let us introduce our notation. An expression of the form F1 F2will mean that there exists C > 0 such that C1F1 F2 CF1; moreover, theconstant C will, as a rule, be independent of all or some of the variables of thefunctions F1 and F2. If 1 p and f = f() is a random variable on aprobability space (, ,P), then

fp = (E|f|p)1/p =

|f()|p dP()1/p;while ifa = (an)n=1 is a number sequence, then

ap = n=1

|an|p1/p

(the formulae must be modified in the natural way for p = ). As usual, thespace Lp consists of all random variables f such that fp < , and lp consists ofall sequences a = (an)

n=1 such that ap < .

We denote by

|E

|the Lebesgue measure of the subset E of [0, 1].

1. Petre KKK-functional and real interpolation method

Let (X0, X1) be a Banach couple, let x X0 + X1, and assume that t > 0. Weconsider the Petre K-functional

K(t, x; X0, X1) = infx0X0 + tx1X1 : x = x0 + x1, x0 X0, x1 X1,

which plays an important role in operator interpolation theory.It is easy to show that for fixed x X0 + X1 the K-functional is an increasing

continuous concave function of t (see [11], 3.1).In what follows we require only the functional K1,2(t, a) = K(t, a; l1, l2) con-

structed for the Banach couple (l1, l2), two approximations of which are discussed

in the lemmas below. The first lemma, due to Montgomery-Smith (see [12]), willbe essential in the proof of Theorem 4. For this reason we present its proof herefor completeness.


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For arbitrary a = (an)n=1 and t N we define the norm

aQ(t) = sup tj=1

nAj

a2n

1/2, (1.1)

where the supremum is taken over all partitionings {Aj}tj=1 of the set of positiveintegers

N.

Lemma 1. If a = (an)n=1 l2 and t2 N, thenaQ(t2) K1,2(t, a)

2aQ(t2). (1.2)

Proof. It follows from the definition ofQ(t) that

aQ(t2) a1 and aQ(t2) ta2.Hence

K1,2(t, a) = inf{b1 + tc2 : b + c = a, b l1, c l2} inf{bQ(t2) + cQ(t2) : b + c = a, b l1, c l2} aQ(t2)

which proves the left-hand side of (1.2).To prove the right-hand side of (1.2) we note first of all that

K1,2(t, a) = sup

n=1

anbn : b = (bn)n=1 l2, J,2(t1, b) 1

,

where J,2(s, b) = max{b; sb2} (see [11], 2.7).Hence, for each > 0 there exists a sequence b l2 such that

(1 )K1,2(t, a) n=1

anbn and J,2(t1, b) = 1.

We select n0, n1, n2, . . . , nt2 {0, 1, . . . , } by induction as follows: if we havealready selected n0, n1,..., nm, where 0 = n0 < n1 < < nm, then

nm+1 = 1 + sup

k :k

n=nm+1

b2n 1

.

Since b 1, it follows thatnm+1

n=nm+1b2n 2. In view of the inequality b2 t,

we have nt2 = .Hence

(1 )K1,2(t, a) n=1

anbn

t2m=1

nmn=nm1+1

b2n

1/2 nmn=nm1+1

a2n

1/2

2 aQ(t2).

The last relation holds for all > 0, which proves the lemma.

Another relation, Holmstedts formula is well known (see [13] or [11], 5.7).


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784 S. V. Astashkin

Lemma 2. There exists a constant > 0 independent of a = (an)n=1 l2 andt > 0

such that

1 [t2]i=1

ai + t

i=[t2]+1

(ai )2

1/2 K1,2(t, a)

[t2]i=1

ai + t

i=[t2]+1

(ai )2

1/2,

(1.3)where (ai )i=1 is a decreasing rearrangement of the sequence (|an|)n=1 and [z] is the

integer part of the quantity z.

Corollary 1. The function (t, a) = K1,2(

t, a) is continuous and increases fort 0. For each sequence a = (an)

n=1 l2,

(1) limt0+ (t, a) = 0;(2) 1a1 limt+ (t, a) a1.

To conclude the section we recall the definition of the real interpolation method.For an arbitrary Banach couple (X0, X1), 0 < < 1, 1 q , we consider thenorm

x,q =

n=

2nK(2n, x; X0, X1)q1/q

, x X0 + X1

(and its natural modification for q = ). The spaces

(X0, X1),q = {x X0 + X1 : x,q < }

are interpolation spaces for the couple (X0, X1) (that is, each linear operatorbounded in X0 and X1 is bounded also in (X0, X1),q); they are called the spacesof the real interpolation method.

2. Systems of random variables equivalent

by distribution to the Rademacher system

Theorem 1. Let{fn}n=1 be a sequence of random variables on a probability space(, ,P). Then the following conditions are equivalent:

(1) {fn} P {rn} ({rn}n=1 is the Rademacher system on [0, 1]);(2) the equivalence

mn=1

anfn

t

mn=1

anrn

t

holds with constant independent of t [1, ), m N, and an R (n =1, . . . , m);

(3) the equivalence

n=1anfnt K1,2(

t, a) (2.1)

holds with constant independent of t [1, ) and a = (an)n=1 l2.


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Proof. As shown in [14], with constant independent oft [1, )and a = (an)n=1 l2we have

n=1

anrn

t

K1,2(

t, a) (2.2)

(this relation is already implicit in [12]). Hence the equivalence (2) (3) is obvious.The implication (1) (2) is a consequence of the definition of systems equivalentin distribution, therefore it remains to establish the implication (3) (1).

We fix a = (an)mn=1, m N (where not all the an are equal to zero) and showthat the distribution of the absolute value of the random variable

f() =mn=1

anfn() ( )

is determined by the behaviour of its moments ft and therefore, by assumption,by the function (t, a) = K1,2(

t, a) (t 1).

Let 1 be the constant involved in the equivalence (2.1). In view of the PaleyZygmund inequality [15], 1.6 and the concavity of the K-functional for t 1, weobtain

P{|f()| (2)1(t, a)} P{|f()|t 2tftt}

(1 2t)2 f2tt

f2t2t 4t(1 2t)2

(t, a)

(2t, a)

2t (2)4t.

The function (t, a) increases in t, therefore

P{|f()| (2)1(t, a)} (2)4t4 (2.3)

also for all t > 0.Conversely, by Chebyshevs inequality and assumption (2.1) we obtain

P{|f()| (t, a)} t ft

(t, a)

t

t

for arbitrary t 1 and > 0. In particular, if = e, then

P{|f()| e(t, a)} et (t 1)

or

P{|f()| e(t, a)} e1t (t > 0). (2.4)

Following [14] we now define the functionals

F(s) = sup{t > 0 : (t, a) s}, G(s) = inf{t > 0 : (t, a) s}.


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786 S. V. Astashkin

If 0 < z < (4)1a1, then, by Corollary 1, there exists t > 0 such that(t, a) 4z. It then follows from (2.3) that

P{|f()| > z} P{|f()| (2)1(t, a)} (2)4t4.Hence, by the definition of the functional G,

P{|f()| > z} (2)4G(4z)4,so that

P{|f()| > z} C11 eC2G(4z) (z < (4)1a1), (2.5)where C1 = (2)

4 and C2 = 4 ln(2).Setting C3 = 4

2 C2 we claim that

C2G(4z) F(C3z) (z > 0).

For, in view of the concavity of the K-functional and the definition of G,

(C2G(4z), a)

2 C2(21G(4z), a) 4

2 C2z = C3z,

and it remains to use the definition of the functional F. Thus, (2.5) can be written

as follows:

P{|f()| > z} C11 eF(C3z) (z < (4)1a1). (2.6)Note that by (2.1) we obtain

fn = limt+

fnt ,

so that f a1. HenceP{|f()| > z} = 0 (z a1). (2.7)

Further, by Corollary 1 we can find t > 0 such that 0 < (t, a) (2e)1z. Itnow follows from (2.4) that

P{|f()| > z} P{|f()| e(t, a)} e1t,so that, by the definition of F,

P{|f()| > z} e1F(C14 z) (z > 0), (2.8)where C4 = 2e.

In view of (2.2), relations similar to (2.6)(2.8) hold also for the Rademachersystem. Namely, if 1 is the constant involved in the equivalence (2.2), thenthe function r(x) =

mn=1 anrn(x) satisfies the relations

|{|r(x)| > z}| (C1)1eF(C

3z) (z < (4)1a1), (2.6)

|{|r(x)| > z}| = 0 (z a1), (2.7)|{|r(x)| > z}| e1F((C4)1z) (z > 0). (2.8)


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LetA = max(C1e; C3C

4; C

1e; C

3C4; 4

). (2.9)

Note that, since the constants and are universal, the dependence of A on thesystem {fn}n=1 reduces to its dependence on , the constant involved in (2.1).

If z < (4)1a1, then relations (2.8) and (2.6) and our choice of A showthat

|{|r(x)| > Az}| e1F((C4)1Az) e1F(C3z) C1eP{|f()| > z} AP{|f()| > z}. (2.10)

On the other hand if z (4)1a1, then it follows from (2.9) that Az a1,so that |{|r(x)| > Az}| = 0 in view of (2.7). Hence inequality (2.10) also holds inthat case.

Conversely, if z/A < (4)1a1, then it follows from (2.8), (2.6), and ourchoice of A that

P{|f()| > z} e1F(C14 z) e1F(C3(z/A))

C1e |r(x)| >z

A A |r(x)| >z

A . (2.11)If z/A (4)1a1, then by (2.9) we obtain the inequality z a1. HenceP{|f()| > z} = 0 in view of (2.7), and (2.11) holds again.

Inequalities (2.10) and (2.11) mean that {fn} P {rn}, which completes the proofof Theorem 1.

Remark 1. Of course, a similar result holds also for finite collections of random vari-ables. Moreover, it is clear from the proof that the constants involved in the equiva-lences in conditions (1)(3) of Theorem 1 depend only on one another. For instance,if (2.1) holds for collections {fn}Nn=1 with constant independent of N = 1, 2, . . . ,then {fn}Nn=1 P {rn}Nn=1 also with constant independent of N = 1, 2, . . . .

We now apply the above theorem to two more special cases.

Recall that a system of random variables {fn}n=1 is said to be multiplicative iffor arbitrary, pairwise distinct n1, n2, . . . , nk (k N) we have

E(fn1fn2 fnk) = 0.

If, in addition,E(fn1fn2 fnk f2n) = 0

for arbitrary, pairwise distinct n1, n2, . . . , nk and n = ns (s = 1, 2, . . . , k), then wesay that the system {fn} is strongly multiplicative. In the case of finite collectionsthe definitions are completely similar.

We now discuss the most important examples of such systems [16].

Example 1. Let fn(x) = sin(2knx) (x

[0, 1]). If kn+1/kn 2, then this is

a multiplicative system and if kn+1/kn 3, then it is a strongly multiplicativesystem.


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788 S. V. Astashkin

Example 2. A sequence {fn}n=1 of independent random variables such thatfn L2 and E(fn) = 0 (n = 1, 2, . . . ) is a strongly multiplicative system.

Kwapien and Jakubowski have established the following result [16].

Theorem D. Let{n}Nn=1 be a multiplicative and {n}Nn=1 a strongly multiplica-tive system of random variables. If for each n = 1, . . . , N ,

n n E(2n), (2.12)then there exist a probability space (, ,P), a-subalgebra0 of the -algebra ,and a random vector (1,

2, . . . ,

N) on (

, ,P) such that this vector itself isequidistributed with (1, 2, . . . , N), while (1, 2, . . . , N) is equidistributed withthe conditional expectations E

(1,

2, . . . ,

N) | 0

.

In particular, for an arbitrary convex function H: Rn R,E

H(1, . . . , N) E

H(1, . . . , N)

.

From Theorems D and 1 we obtain the following result.

Theorem 2. Let{fn}n=1 be a strongly multiplicative system of random variablessuch that |fn()| D (n = 1, 2, . . . ; ) and d = infn=1,2,...E(f2n) > 0. Then

{fn}P

{rn}, and the constant involved in the equivalence depends only on D and d.Proof. Assumption (2.12) of Theorem D holds for the systems n = fn/D andn = rn. Hence for all t 1, m N, and an R (n = 1, 2, . . . , m) we have

mn=1

anfn

t

D

mn=1

anrn

t

.

Conversely, if n = (d/D)rn, n = fn, then (2.12) holds again, so thatmn=1

anrn

t

D

d

mn=1

anfn

t

.

To complete the proof it remains to use Theorem 1.

Corollary 2. Each sequence of independent random variables {fn}n=1 such that

fn L2, E(fn) = 0 , |fn()|D (n = 1, 2, . . . ; ), and d = infn=1,2,...E(f2n) > 0is equivalent in distribution to the Rademacher system.

We consider now another situation. Let G be a compact Abelian group, thegroup of its characters, and the Haar measure on G. In accordance with theabove definition a subset F of is called a Sidon set if

|f()| Cffor some C = C(F) and each f C(G) such that

f() =

G

f d = 0 ( / F).

Pisier [17] has proved the following result (we present its formulation only in thescalar case).


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Theorem E. If F = {n} is a Sidon set, then the equivalence

mn=1

ann

t

mn=1

anrn

t

(t 1)

holds with constant depending only on the Sidon constant C(F).

Theorems E and 1 immediately give us the following result, which was alsoproved (by another method, but also on the basis of Theorem E) in [18].

Theorem 3. Each infinite Sidon system F = {n}n=1 of characters of a com-pact Abelian group is equivalent in distribution to the Rademacher system. Thecorresponding equivalence constant depends only on the Sidon constant C(F).

Corollary 3. Sequences

fn(x) = sin(2knx) and gn(x) = cos(2knx) (x [0, 1])

in whichkn+1/kn > 1 are equivalent in distribution to the Rademacher system.

Remark 2. We point out the fact that it was the similarity between the behaviourof Rademacher series and lacunary trigonometric series (see the introduction) thatwas the starting point of the study of deeper connections between these seriesand led eventually to the investigation of systems equivalent in distribution to themodel Rademacher system. Among papers considering these subjects, besides[16][18], we also mention a note [19] by Rodin and Belov, where the authors usedTheorem D to carry over the results of [20] on the behaviour of Rademacher seriesin symmetric function spaces to lacunary trigonometric series. As mentioned in thatnote, in 1988 one of its authors reported at the North-Caucasian workshop that inmany symmetric spaces the norms of polynomials in the Rademacher system andof lacunary trigonometric polynomials are the same up to equivalence. Corollary 3shows that this actually holds in all symmetric spaces, not just in many, becausethe norm of a function in such a space is defined in terms of the distribution of itsabsolute value (see [21], 2.4). This allows us, in particular, to improve some resultsof [19]. For instance, in Theorem 2 from that paper, which establishes necessaryand sufficient conditions for the equivalence of the norms of lacunary trigonometricpolynomials in a symmetric space to the norms of the sequences of their coefficientsin the spaces l1 and l2, the assumption that the symmetric space is an interpolationspace for the couple (L1, L) is superfluous.

3. Selection of subsystems equivalent in

distribution to the Rademacher system

Theorem 4. Let{fn}n=1 be a system of random variables on a probability space(, ,P) containing a subsequence {fnk}k=1 such that

(1) |fnk()| D (k = 1, 2, . . . ; );(2) fnk 0 weakly in L2;(3) d = infk=1,2,... fnk2 > 0.


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790 S. V. Astashkin

Then there exists a subsystem {i}i=1 {fn}n=1 such thati=1

aii

t

K1,2(

t, a) (a = (ai)i=1 l2, t 1) (3.1)

with constant dependent only on D and d.

Proof. As shown by Gaposhkin (see [2], Theorem 1.3.2, a full proof can be foundin [1]), if conditions (1) and (2) are satisfied, then one can select a subsequence{gi} {fnk} such that for some C1 = C1(D) and all a = (ai)i=1 l2 and t 1 wehave the inequality

i=1

aigi

t

C1

t a2. (3.2)

Hence, for each expansion (ai) = (bi) + (ci) with (bi) l1 and (ci) l2 we have

i=1

aigi

t

i=1

bigi

+

i=1

cigi

t

Di=1

|bi| + C1t i=1

c2i1/2 max(C1; D) b1 + t c2.Consequently, by the definition of the K-functional we obtain

i=1

aigi

t

max(C1; D)K1,2(

t, a) (a = (ai)i=1 l2, t 1). (3.3)

In the proof of the reverse inequality we use the equivalence of the K-functionalK1,2(t, a) and the norm Q(t2) (see (1.1) and (1.2)), and also the upper estimateof the Lq-norms (q > 1) of modified Riesz products (for applications of classicalproducts in similar cases, see [2], 1.4 or [10], 8.4).

First, assumptions (1) and (3) of the theorem show that 0 < d gi2 D(i = 1, 2, . . . ), therefore, in view of the positive homogeneity of both sides of (3.1)we can assume that gi2 = 1 (i = 1, 2, . . .).

Let (i)i=1 be a number sequence such that

i 0, 0 < i < 116

min(1; D),

k=i+1

k < i (i = 1, 2, . . . ). (3.4)

By the assumptions of the theorem the sequence g2i is weakly compact in L2.Hence there exists {hk} {gi} such that h2k h weakly in this space, where0 h() D2 and E(h) = 1. Together with {gi}, the sequence ofhk also convergesto 0 weakly in L2. Hence we can find an index k1 such that the function 1 = hk1satisfies the relation

|E(1)| + |E(h1)| + |E(21 h)| 1

2D.


14/30


Assume that we have already selected the indices k1 < k2 < < ki1 and thefunctions 1 = hk1 , 2 = hk2, . . . , i1 = hki1 (i 2). We now find ki > ki1such that for i = hki we have the relation

|E(j1 jsi)| + |E(hj1 jsi)| + |E[j1 js(2i h)]|

+s

l=1

|E(j1 jl12jljl+1 jsi)| 2iD1i, (3.5)where j0 = js+1 = 1 and the summation proceeds over all collections of indicessuch that 1 j1 < j2 < < jl < < js i 1 (s = 1, 2, . . . , i 1).

We claim that the sequence {i}i=1 satisfies (3.1) for arbitrary t 1.Let t N first, and let {Aj}tj=1 be an arbitrary partitioning of the positive

integers into t sets. For N N and j = 1, 2, . . . , t let ANj = {i = 1, . . . , N : i Aj}(some of these sets may be empty). We consider now the Riesz products formedby blocks corresponding to the sets ANj :

RN() =N

i=1(1 + bii()) =

t

j=1 iANj(1 + bii()),

where the bi are real coefficients such thatiAj

b2i D2. (3.6)

Reasoning along the lines of Sidons classical method, for an arbitrary function

() =i=1

aii(), a = (ai)i=1 l2,

we find a lower estimate of the integral

IN = RN()() dP() =

i=1 aiE(RNi) =

i=1 aii,N. (3.7)Here

i,N = E(RNi) =

1 +

Nk=1

bkk() +

1k1 N), and

i,N = E(i) +N

k=1,k=i

bkE(ki) +

1k1


15/30

792 S. V. Astashkin

Theni,N = S

i1 + S

i2 + S

i3, (3.9)

where the sums Sik (k = 1, 2, 3) are defined as follows. Setting

E(k1 ks2i ) = E(k1 ksh) + E[k1 ks(2i h)],

we collect in Si1 all terms containing integrals of the form E(k1 ksi) orE[k1 ks(2i h)], where k1 < k2 < < ks < i. The terms containing inte-grals E(k1 kl1ikl ks), where k1 < k2 < < kl1 i < kl < < ks,1 l s, we collect in Si2, and in S

i3 we include the terms containing integrals

E(k1 ksh).In view of (3.5), (3.4), and (3.6) we obtain the estimates

|Si1| iD

, |Si2| 1

D

k=i+1

k N there areno sums Si2 and S

i3 in (3.9), therefore

i=N+1

|i,N| 1D

i=N+1

i 1

16D. (3.12)

Assume now that 1 i N. For each j = 1, 2, . . . , t, in view of (3.9)(3.11),(3.4), (3.6), we have

iAN

j

2i,N1/2

3k=1

iAN

j

(Sik)21/2

2

D

iAN

j

i +1

8

iAN

j

b2i

1/2

3

8D. (3.13)

Relations (3.7) and (3.8) show that

IN =Ni=1

aibi +Ni=1

aii,N +

i=N+1

aii,N

=t

j=1

iAN

j

aibi + tj=1

iAN

j

aii,N+ i=N+1

aii,N.


16/30


For each j = 1, 2, . . . , t we now select bi (i ANj ) such that (3.6) holds and

iANj

aibi =1

D

iANj

a2i

1/2;

we also select N N such that

a2 2 Ni=1

a2i

1/2 2

tj=1

iAN

j

a2i

1/2.

Then it follows by the above equality and (3.12), (3.13), and (3.4) that

IN 1

D

tj=1

iAN

j

a2i

1/2

tj=1

iAN

j

a2i

1/2iAN

j

2i,N

1/2

i=N+1

|ai| |i,N|

5

8D

t

j=1iANj a2i

1/2

i=N+1 |ai|2

1/2

i=N+1 2i,N

1/2

1

2D

tj=1

iANj

a2i

1/2.

The last inequality shows that for each t N and an arbitrary partitioning {Aj}tj=1of positive integers there exist sufficiently large N and coefficients bi (i = 1, 2, . . . , N )satisfying (3.6) such that

IN =

RN()() dP() 1

3D

tj=1

iAj

a2i

1/2. (3.14)

The next part of the proof is the estimate of the integral IN by an expression ofthe form Ct, with constant C > 0. By Holders inequality

|IN| RNtt, t = tt 1 , (3.15)

and the non-negativity of Riesz products it is sufficient for this to find an estimateof the quantity

LN = RNtt =

[RN()]t dP() (3.16)

(for the proof that {i}i=1 is a Sidon system we require only the estimate of theL1

-norms of Riesz products (see [2], Theorem 1.4.1)).We consider t N, t 3. As before, let {Aj}tj=1 be a partitioning ofN, let

ANj = {i = 1, . . . , N : i Aj}, and let bi be coefficients satisfying (3.6).


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794 S. V. Astashkin

By the elementary inequality (1 + x)y 1 + yx (x 1, 0 < y 1),

[RN()]t

Ni=1

(1 + bii())[1 + (t 1)1bii()]

N

i=11 +

D2

t 1b2i +

t

t 1bii()

=t

j=1

iAN

j

1 +

D2

t 1 b2i +

t

t 1 bii()

. (3.17)

Let m(B) be the cardinality of a subset B ofN. Then the inner product in the lastexpression is equal to the sum

1 +D2

t 1iANj

b2i +

D2

t 12

i1


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Moreover, since the sets ANj (j = 1, 2, . . . , t) are pairwise disjoint, the inte-grands in the last expression are distinct products of pairwise distinct functions i(i = 1, 2, . . . , N ). The number of these functions is m(Cj) (j = 1, 2, . . . , t) for termsin the first sum, m(Cj1 )+ m(Cj2) (1 j1 < j2 t) for terms in the second, . . . , andt

j=1 m(Cj) for terms in the last sum. Hence the largest index of functions i in

each integrand is at least m(Cj), m(Cj1 ) + m(Cj2), . . . , tj=1 m(Cj), respectively.

As a result, from (3.18) and (3.19) we deduce

LN 4e

Ni=1

2i

1j1


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796 S. V. Astashkin

Relation (3.2) shows that {gi}i=1, and therefore also {i}i=1, is an Sp-systemfor each p < . Hence it is a Banach system ([2], Corollary 1.3.1), that is, for someM = M(D) we have

a2 M

mi=1

aii

1

(a = (ai)i=1 l2). (3.24)

Since a2 a1, it follows that K(1, a; l1, l2) = a2. By the concavity of theK-functional we can now deduce from (3.23) and (3.24) that

K1,2(

2, a)

2K1,2(1, a) 22 (3.25)and

K1,2(1, a) M1. (3.26)For arbitrary t 1 we can find a positive integer t0 such that t0 t < t0 + 1.

Now, in view of the concavity of the K-functional again and also by (3.21), (3.25),and (3.26) we obtain

K1,2(

t, a)

t/t0K1,2(

t0, a)

2K1,2(

t0, a) Ct,where C =

2 max(2; M; 6

2 D) depends only on D.

Thus, relation (3.3) holds also for the system

{i

}i=1, which completes the proof

of the theorem.The next two results are immediate consequences of Theorems 1 and 4.

Theorem 5. From a system {fn}n=1 of random variables on a probability space(, ,P) one can select a subsystem {i}i=1 equivalent in distribution to theRademacher system on [0, 1] if and only if there exists a subsequence {fnk} of {fn}such that

(1) |fnk()| D (k = 1, 2, . . . ; );(2) fnk 0 weakly in L2;(3) d = infk=1,2,... fnk2 > 0.

Moreover, the constant involved in the equivalence will depend only on D and d.

Proof. By Theorem 4 we can select a subsequence {i}i=1 {fn}n=1 such thatthe equivalence (3.1) holds. Then, however, {i}

P

{rn} by Theorem 1 and theproof of the sufficiency is complete.

The necessity of conditions (1) and (3) is a consequence of the definition of equiv-alence in distribution and the fact that the Rademacher system has these properties.As pointed out in the introduction, the Rademacher system is an Sp-system for each

p < . Hence a system equivalent to it in distribution must also have this property.Thus, the necessity of condition (2) is a consequence of Theorem A stated in theintroduction.

Remark 3. In connection with Theorem 5 we recall a problem posed by Alexits[22], 3.2. Is it always possible to select an infinite multiplicative or even strictlymultiplicative subsystem from an arbitrary complete orthogonal system of func-tions? The result of Theorem 5 answers (in the affirmative) a closely related ques-

tion: from an arbitrary system of random variables with properties (1)(3) one canselect a subsystem that is equivalent in distribution to the strictly multiplicative(more than that: consisting of independent functions) Rademacher system.


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Theorem 6. If {fn}n=1 is an orthonormal system of random variables on a prob-ability space (, ,P), |fn()| D (n = 1, 2, . . .; ), then one can select fromthis system a subsequence {i}i=1 equivalent in distribution to the Rademachersystem. The constant involved in the equivalence depends only on D.

It is clear that a sequence {i}i=1 satisfying (3.1) is a Sidon system (and alsoan Sp-system for all p 0 dependent only on Dand d and each polynomial

T() =mi=1

aii()

there exists a set E= E(T) , P(E) > 12m, such that

|T()| 2T 3mi=1

|ai| ( E).

Proof. By Theorem 5 there exists a subsequence {i} {fn} such that for someC > 0 and all z > 0 we have the inequality

C1{|T(x)| > C z} P{|T()| > z} C{|T(x)| > C1z},

where T(x) = mi=1 airi(x) (x [0, 1]). In particular, this shows thatC1T T CT.

Hence it follows by the definition of the Rademacher system that

P{|T()| > (2C2)1T} P{|T()| > (2C)1T}

C1{|T(x)| > 21T} C12m.We set 1 = C1, 2 = (2C2)1, and E = E(T) = { : |T()| > 2T}.

Then for E we have

|T()| > 2T 2C1 T = 3 mi=1

|ai|,

where 3 = 2C1.

As already pointed out in the introduction (see Theorem B), under the sameassumptions about a sequence of measurable functions on [0, 1] as in Theorem Bone can select a subsystem of this sequence that has the SidonZygmund property

and, under the additional assumption (0.3), the SidonZygmund property in thenarrow sense. The following two theorems show that even somewhat stronger resultsactually hold.


21/30

798 S. V. Astashkin

Theorem 7. Let {fn}n=1 be a sequence of measurable functions on [0, 1] thatcontains a subsequence {fnk} with properties (1)(3) in Theorem 5. Then thereexists a subsystem {i} {fn} with the following property.

For some subset E of [0, 1], |E| > 0, and each interval I [0, 1], |I E| > 0,there exists an index k0 = k0(I) such that the sequence {iI}i=k0 (I(x) = 1,x I, and I(x) = 0, x / I) is equivalent in distribution to the Rademachersystem on [0, 1]. The constant involved in this equivalence depends only on D, d,and the interval I.

Proof. By Theorem 1 it is sufficient to find a subsystem {i} of{fn} and a subset Eof [0, 1], |E| > 0, such that for each interval I [0, 1], |I E| > 0, there exists anindex k0 = k0(I) such that

i=1

aik0+i1I

t

K1,2(

t, a) (a = (ai)i=1, t 1)

(the constant involved in this equivalence depends on D, d, and I).The arguments bringing us to this relation are mostly similar to the ones used

in the proof of Theorem 4. We merely make several observations.

First of all, inequality (3.2) remains valid, of course, if we replace the functionsgi by giI (here I is an arbitrary subinterval of [0, 1]).Next, as in the proof of Theorem 1.4.2 in [2], one can select a subsequence

{i} {gi} such that relations of type (3.5) hold for integrals over each intervalI [0, 1].

Finally, the well-known Marcinkiewiczs lemma ([2], Lemma 1.2.5) allows us toassume that there exists a subset E of [0, 1], |E| > 0, such that for each F E,|F| > 0, the inequality

lim infi

F

2i (x) dx > 0 (3.27)

holds. After that one virtually repeats the proof of Theorem 4. The only differenceis that one must replace integrals over [0, 1] by integrals over an interval I such that

|I

E

|> 0.

Theorem 8. Let {fn}n=1 be a sequence of measurable functions on [0, 1] thatcontains a subsequence {fnk} satisfying conditions (1) and (2) of Theorem 5 andalso condition (0.3).

Then there exists a subsystem {i} of {fn} such that for each interval I [0, 1],|I| > 0, there exists an index k0 = k0(I) such that the sequence {iI}i=k0 isequivalent in distribution to the Rademacher system on [0, 1].

The proof of Theorem 8 is perfectly similar to the proof of the preceding theo-rem because, in view of (0.3), we can assume that the selected subsystem {i}i=1satisfies (3.27) for each set F [0, 1] of positive measure.

4. Density of subsystems equivalent in

distribution to the Rademacher system

We consider now the problem of the selection of finite subsystems of uniformlybounded orthonormal systems of functions {fn}Nn=1 on [0, 1]. We claim that such a


22/30


system always contains a subsystem {fni}si=1 of logarithmic density (s Clog2 N)that is equivalent in distribution to the system of the first s Rademacher functionswith constant independent of s. This is an improvement on Theorem C in theintroduction established by Kashin [9].

Theorem 9. Let{fn}Nn=1 be an orthonormal system of functions on [0, 1] such that

|fn(x)

| D (n = 1, 2, . . . , N ). Then it contains a collection of functions

{fni

}si=1,

where s max{[ 16 log2 N]; 1} and 1 n1 < n2 < < ns N, such that for someC > 0 dependent only on D and all ai R (i = 1, 2, . . . , s) and z > 0 the followinginequality holds:

C1

si=1

airi(x)

> Cz

si=1

aifni(x)

> z C

si=1

airi(x)

> C1z.

We require two lemmas for the proof. The first is well known ([10], Lemma 8.4.1).

Lemma 3. Each system of functions {fn}Nn=1 (log2 N 6) satisfying the assump-tions of Theorem 9 contains functions {fni}si=1 (1 n1 < n2 < < ns N),where s [ 16 log2 N], such that

As

E

si=1

fni(x)D

i2 10s. (4.1)

Here As is the set of collections = (i)si=1 such that

(a) i = 0, 1 or 2; (b)

i:i=1

1 1; (c)

i: i=2

1 1.

The second lemma enables one to extend under certain conditions a uniformlybounded system of functions on [0, 1] to a larger set so that it becomes a multiplica-tive system remaining at the same time uniformly bounded. For similar results onthe extension to an orthonormal collection, see [10],

7.1.

Lemma 4. Let{gi}si=1 be a collection of functions on [0, 1] such that |gi(x)| D(i = 1, 2, . . . , s; x [0, 1]) and assume that

maxAs

E si=1

gi(x)

D

i < 2s, (4.2)where As is the set of all collections = (i)

si=1 As such that i = 0 or 1.

Then there exists a multiplicative system {hi}si=1 of functions on [0, 2] such thathi(x)[0,1](x) = gi(x) and hiL[0,2] D (i = 1, 2, . . . , s).Proof. We introduce the partitioning

[1, 2) =2s

k=1

k, k = [ak1, ak), ak = 1 + k2s (k = 1, 2, . . . , 2s),


23/30

800 S. V. Astashkin

and consider some (arbitrary) one-to-one correspondence between the set As and

the family of intervals {k}2s1k=1 . We shall define the functions hi on each intervalk in this family.

Let = (i)si=1 be an element ofA

s corresponding to k. We consider the set

{ij}mj=1 = {i = 1, 2, . . . , s : i = 1}, i1 < i2 < < im (m = 1, 2, . . . , s).For k let

u(x) = [ak1,)(x) [,ak](x) (x k).Then the quantity

v() =

k

u(x) dx = 2 ak1 ak (ak1 ak)

ranges over the entire interval [2s, 2s]. By (4.2) the functions gi(x) = gi(x)/Dsatisfy the inequality |E(gi1 gi2 gim)| < 2s, therefore there exists k (ak1, ak)such that

v(k) = E(gi1 gi2 gim). (4.3)Assume first that m 2 and consider a family of piecewise constant functions

dj(x) (x

[ak1, k) and j = 1, 2, . . . , m

1) such that

(1) |dj(x)| 1;

(2)

kak1

dj1dj2 djl dx = 0 (1 j1 < j2 < < jl m 1)(4.4)

(one can construct such a family, for instance, by carrying over the Rademacherfunctions to [ak1, k) from [0, 1]). Let cj(x) (x [k, ak] and j = 1, 2, . . . , m 1)be a similar system on the interval [k, ak].

Ifx k, thenhi(x) 0 (i = ij, j = 1, 2, . . . , m),

hij(x) = D[dj(x)[ak1,k)(x) + cj(x)[k,ak](x)] (j = 1, 2, . . . , m 1)

and

him(x) = D

m1j=1

dj(x)[ak1,k)(x) m1j=1

cj(x)[k,ak](x)

.

By (4.4) and (4.3),k

hi1hi2 him dx = Dmk

uk(x) dx

= Dmv(k) = E(gi1gi2 gim). (4.5)At the same time, for any other collection of indices 1 i1 < < ik s it followsfrom the definition of the functions hi (i = 1, 2, . . . , s) and from (4.4) that

k

hi1

hi2 hi

ldx = 0. (4.6)


24/30


On the other hand, ifm = 1, then we set

hi(x) 0 (i = i1), hi1(x) = Duk(x),

and relations (4.5) and (4.6) again hold.We now define the functions hi(x) on the entire interval [0, 2] by setting hi(x) =

gi(x) for x

[0, 1] and h

i(x) = 0 for x

2s. In view of (4.5) and (4.6),

{hi}

s

i=1is

a multiplicative system on [0, 2] and |hi(x)| D.Proof of Theorem 9. In view of Theorem 1 and Remark 1 after it, it suffices toshow that there exists a collection {fni}si=1 (s max{[ 16 log2 N]; 1}) such that forsome constant dependent only on D,

si=1

aifni

t

K1,2(

t, a) (t 1, a = (ai)si=1), (4.7)

where, as before, K1,2(

t, a) = K(

t, a; l1, l2).Using Lemma 3 we select a subsystem {fni}si=1 (s max{[ 16 log2 N]; 1}) satis-

fying (4.1). Since As As, the functions gi = fni satisfy at the same time (4.2).Hence, by Lemma 4, we can extend the functions

{fn

i}s

i=1to a multiplicative sys-

tem {hi}si=1 on [0, 2], |hi(x)| D. Then, however, the functions hi(x) = hi(2x)make up a multiplicative system on [0, 1], |hi(x)| D, and therefore by Corollary 3in [16], for arbitrary t 1 and a = (ai)si=1 we obtain

si=1

aihi

Lt[0,1]

C1

t a2,

where C1 > 0 depends only on D. Hences

i=1

aifni

t

2C1

t a2, (4.8)

which in a similar way to the proof of Theorem 4 (see inequality (3.20)) shows thats

i=1

aifni

t

CK1,2(

t, a) (t 1), (4.9)

where C = max(D; 2C1) depends only on D.For the proof of the reverse inequality we set

a = (ai)si=1 and P(x) =

si=1

aifni(x).

Let t N first, t 3. Ifs 16D2, then by the definition of the K-functional wehave

Pt P2 = a2 1s

si=1

|ai| 14D

K1,2(

t, a). (4.10)


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802 S. V. Astashkin

Hence it suffices to consider the case when

s > 16D2. (4.11)

We consider now the Riesz products

Rs(x) =s

i=1

1 + biD

fni(x).For an arbitrary partitioning {Aj}tj=1 of the set {1, 2, . . . , s} into t disjoint subsets(some of which may be empty) we assume that

iAj

b2i 1 (j = 1, 2, . . . , t). (4.12)

We now find a lower estimate for the integral

Is = 1

0

P(x)Rs(x) dx

=s

i=1

ai

10

fni(x) dx +s

i=1

ai

10

fni(x)As

sk=1

bkD

fnk(x)

kdx.

Rearranging the terms we obtain

Is = D1

si=1

aibi +s

i=1

aii, (4.13)

where

i = 1

0fni (x) dx +

1

0fni(x)

As(i)

s

k=1

bkD fnk(x)k

dx,

As(i) = As \ {i}, i = (ij), ii = 1, ij = 0 (j = i).

In view of (4.1) and the condition |bi| 1 (i = 1, 2, . . . , s), after applying theCauchySchwarzBunyakovski inequality we obtain

si=1

|i| DAs

10

sk=1

fnk(x)

D

kdx

D(s2s)1/210s/2 = D(s5s)1/2.In view of (4.11), (s5s)1/2 8/s < 21D2, therefore

si=1

|i| 12D

. (4.14)


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For each j = 1, 2, . . . , t we now find bi (i Aj) such that (4.12) holds andiAj

aibi =

iAj

a2i

1/2.

Then by (4.13) and (4.14) we obtain

Is 1

D

tj=1

iAj

a2i

1/2

tj=1

iAj

aii

1

D

tj=1

iAj

a2i

1/2

tj=1

iAj

a2i

1/2iAj

|i|

1

2D

tj=1

iAj

a2i

1/2.

The partitioning {Aj}tj=1 is arbitrary, therefore by Lemma 1 we finally obtain

Is 12

2DK1,2(t, a) (t N, t 3). (4.15)

Next, we find an upper estimate for Is. By Holders inequality,

|Is| RstPt, t = tt 1 .

As in the proof of Theorem 4, we claim that for t 3 we have

Rst 4e2sAs

E si=1

fni(x)

D

i,therefore by (4.1), the inclusion As

As, and the CauchySchwarzBunyakovski

inequality we obtain the relation Rst 4e. Hence |Is| 4ePt and it followsfrom (4.15) that

K1,2(

t, a) 8

2 eDPt (t N, t 3). (4.16)

Similar relations hold also for t = 2 and t = 1. In fact, {fn}Nn=1 is an orthonormalsystem and K1,2(1, a) = a2, so that

K1,2(

2, a)

2K1,2(1, a) =

2 P2. (4.17)

On the other hand, ift = 1, then, in view of (4.8) and as in the proof of Theorem 4,we can use [2], Corollary 1.3.1, and therefore

K1,2(1, a) = a2 MP1 (4.18)with M > 0 dependent only on D.


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804 S. V. Astashkin

Finally, passing from positive integers (see (4.16)(4.18)) to arbitrary t 1 weobtain in the standard way

K1,2(

t, a) CPt,

where the constant C =

2 max(M;

2; 8

2 eD) depends only on D. Hence, in

view of (4.9), the equivalence (4.7) holds with constant dependent only on D, whichcompletes the proof.

Remark 4. The example of a trigonometric system shows that the result of Theo-rem 9 (similarly to Theorem C in the introduction) is best possible in order. ForStechkin [23] has shown that if a sequence {2 cos2nkx}k=1 (x [0, 1]) is a Sidonsystem, then

k:nk 0we have

mi=1

airni(x)

> z =

mi=1

airi(x)

> z.

5. K-closed representability of Banach couples

We say that a Banach couple (X0, X1) is K-closed representable in a Banachcouple (Y0, Y1) if there exists a linear operator T: X0 + X1 Y0 + Y1 such that

(a) T is a bounded injective operator from X0 into Y0 and from X1 into Y1;(b) for some constant C > 0 independent of x X0 + X1 and t > 0,

K(t , Tx; Y0, Y1) K(t, x; X0, X1).

As shown in [24] (see also [25]), a couple (l1, l2) is K-closed representable in thecouple (L, G) of spaces of measurable functions on [0, 1]. Here we denote by G

the closure of the space L in the Orlicz space LN with N(u) = eu2 1 and we

can define the operator T as follows:

T a(x) =n=1

anrn(x) (a = (an)n=1 l2).

As before, {rn}n=1 is the Rademacher system on [0, 1].Assume that a sequence {fn}

n=1 of measurable functions on [0, 1] has properties(1)(3) in Theorem 5. Then we can select a subsystem {i} of it that is equivalentin distribution to the Rademacher system. All the proofs in [24] use only estimates


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of the distributions of the absolute values of polynomials in the Rademacher system,therefore the corresponding results hold also for {i}. In particular, the operator

Ta(x) =i=1

aii(x) (a = (ai)i=1 l2)

also brings about a K-closed representability of the couple (l1, l2) in (L, G).We now prove a negative result on the K-closed representability of couples, which

is related to the above example.

Theorem 10. The couple (l1, l2) is not K-closed representable in the couple ofspaces (L, L2) of random variables on a probability space (, ,P).

Proof. Assume that, on the contrary, there exists a linear operator T: l2 L2 suchthat

T a a1 (a l1), T a2 a2 (a l2) (5.1)and for some C1 > 0,

C11 K1,2(t, a) K(t , Ta; L, L2) C1K1,2(t, a) (a l2, t > 0), (5.2)

where, as before, K1,2(t, a) = K(t, a; l1, l2).Let fn = T en, where en = (jn),

nn = 1,

jn = 0 (j = n). Then it follows

from (5.1) that there exist D > 0 and d > 0 such that

D1a1 n=1

anfn

Da1 (a l1)

and

da2 n=1

anfn

2

d1a2 (a l2). (5.3)

Hence, in particular,

|fn()| D (n = 1, 2, . . .; ) and fn2 d. (5.4)

Moreover, it follows from (5.3) that fn = 0 (n = 1, 2, . . .) and for all k, m N,k m, we have

kn=1

anfn

2

d2

mn=1

anfn

2

.

Hence {fn}n=1 is a basis in the closed (in the L2-norm) linear hull Z of this system([26], part I, 1). In other words, {fn}n=1 is a Riesz basis sequence, so that forg Z we see that E(gfn) 0 as n [2], 1.1. Ifg L2 is arbitrary, theng = g1 + g2, where g1 Z and g2 Z (Z is the orthogonal complement of Z)and E(gfn) = E(g1fn) 0 as n . Thus,

fn 0 weakly in L2 (as n ). (5.5)


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806 S. V. Astashkin

Relations (5.4) and (5.5) show that the system {fn}n=1 has properties (1)(3) inTheorem 5. Hence there exists a subsequence {i} {fn} equivalent in distributionto the Rademacher system on [0, 1]. Hence, by Khintchines inequality (see [3] or[6], 4.5) we obtain

C12

a2

i=1 aiip C2a2 (a = (ai)i=1

l2), (5.6)

where the constant C2 > 0 depends only on D, d, and p [1, ).At the same time, it follows from (5.2) that

C11 K1,2(t, a) K

t,

i=1

aii; L, L2

C1K1,2(t, a) (a l2, t > 0).

Hence, applying to the couples (L, L2) and (l1, l2) the interpolation functor (,),pwith parameters 0 < < 1, p = 2/ (see 1), in view of [11], Theorem 5.2.1 weobtain

i=1aii

p

a

r,p =

i=1

(ai )pip/r1

1/p,

where r = 2/(2 ) < 2 (the constant involved in the last equivalence dependsonly on C1 and ). Since ar,p a2, the last relation is in contradiction withinequalities (5.6) for p = 2/.

The author is grateful to B. S. Kashin, who posed the problem whose solution isthe subject of 4.

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Samara State University

E-mail address : [email protected]

Received 12/AUG/99Translated by IPS(DoM)

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