J Fourier Anal ApplDOI 10.1007/s00041-014-9332-7

� p-Linear Independence of the System of IntegerTranslates

Ivana Slamic

Received: 18 July 2013 / Revised: 5 March 2014© Springer Science+Business Media New York 2014

Abstract Various properties of the system Bψ of integer translates of a square inte-grable function ψ ∈ L2(R) can be completely described in terms of the periodizationfunction pψ(ξ) = ∑

k∈Z|ψ(ξ + k)|2. In this paper, we consider the problem of �p-

linear independence, where p > 2. The results we present include the method ofconstruction for one type of counterexamples to several naturally taken conjectures,a new sufficient condition for �p-linear independence and a characterization theoremhaving an additional assumption on Bψ . In the latter, we obtain the characterizationin terms of the sets of multiplicity of Lebesgue measure zero.

Keywords Integer translates · �p-Linear independence · Sets of uniqueness

Mathematics Subject Classification 42C15 · 46E30

1 Introduction

It is known (see e.g. [6]) that many properties of the system

Bψ = {ψk : k ∈ Z}, ψk(x) ≡ Tkψ(x) = ψ(x − k)

Communicated by Chris Heil.

I. Slamic (B)Department of Mathematics, University of Rijeka, R.Matejcic 2,51000 Rijeka, Croatiae-mail: islamic@math.uniri.hr

J Fourier Anal Appl

of integer translates of a square integrable function ψ ∈ L2(R) can be expressed interms of the periodization function defined by

pψ(ξ) =∑

k∈Z

|ψ(ξ + k)|2, ξ ∈ R, (1)

where ψ denotes the Fourier transform ofψ . System of translates of a square integrablefunction plays an important role in harmonic analysis since it generates the principalshift invariant space

〈ψ〉 = span{ψk : k ∈ Z}

which is used, for example, in construction of wavelet and Gabor systems. The impor-tance of redundancy in generating systems has already been emphasised in the work ofDaubechies, [3]. Since redundancy in Bψ cannot be obtained considering only finitesums (the system being always linearly independent), one must consider infinite sums,i.e. the various levels of linear independence of infinite type (see e.g. [14]). Our mainpurpose was to find a characterization of �p-linear independence for p �= 2 and wehave presented in [15] a partial result for p < 2 case and a complete characterizationof �1-linear independence. This problem appears as a natural direction in generalizingthe problem of �2-linear independence, which was solved partially in [16] and com-pletely in [13]. Besides this, in the context of sparse representations, it can be usefulto know which spaces the coefficients belong to.

The proof of �2-linear independence and our paper about p < 2 case introducedsome, slightly unexpected, connections with harmonic analysis. During the research onp > 2 case, some results have lead us to several conjectures which, however, appearedto be false. We present here a method of construction for one type of counterexamplesto such statements. Precisely, the conditions we consider are:

pψ > 0 a.e.,1

pψ∈ Lr , for some r < 1 and

1

pψ∈

⋂

r<1

Lr .

On the other hand, we present a new sufficient condition for �p-linear independence,where p > 2, stating that if 1

pψis integrable on intervals such that the complement

of their union in T is a set which is a set of uniqueness for trigonometric series withcoefficients belonging to �p, then Bψ is �p-linearly independent. This weakens thecondition 1

pψ∈ L1(T) which was known as the sufficient condition for ω-linear

independence. We also prove that there exist �p-linearly dependent systems with 1pψ

integrable on intervals such that complement of their union in T is a set of multiplicityof Lebesgue measure zero. Finally, we restrict on a class of functions satisfying certainproperties in order to give a characterization of �p-linear independence in terms ofsets of uniqueness. It is interesting to observe the following thing which might seemconfusing to the reader. Sets of uniqueness for trigonometric series are found amongstsets of Lebesgue measure zero. We consider the periodization functions that are L1(T)-functions, hence our theorems should not distinguish functions that are equal a.e.

J Fourier Anal Appl

The characterization is, however, expressed in terms of the zero set of the specificrepresentative of the class, precisely the one having some analytic properties (e.g.continuity), with the results still holding for the whole class. Precisely, we show that ifthere is a function g, such that

√g satisfies condition

∑∞l=0

∑|k|�l |√g(k)| < ∞, and

g is related to pψ in a sense that gpψ

∈ L1(T) and pψg ∈ L1(T), then Bψ is �p-linearly

independent if and only if the set of zeroes of the continuous version of g is a set ofuniqueness for trigonometric series with coefficients belonging to �p.

The results of this paper are parts of my Ph.D. thesis on redundancy in the systemsof translates, done at the University of Zagreb. I would like to use this opportunity toexpress my gratitude to my advisor, Professor Hrvoje Šikic, for his constant guidanceand support throughout my study.

2 Basic Notions and Tools

Let Cω denote the set of sequences of complex numbers and let K be a subset of C

ω.

Definition 1 Let (H, ‖ · ‖) be a Hilbert space. We shall say that a sequence {xn} in H

is K-linearly independent if

(ck) ∈ K,∞∑

k=1

ck xk = 0 ⇒ ck = 0,∀k ∈ N. (2)

In this paper, we consider the cases where H = L2(R) and K = �p(Z), p > 2,K = c0(Z) or K = C

ω with Z ordered as Z = {0, 1,−1, 2,−2, . . .}. If 1 � p < ∞,then �p(Z) ⊆ c0(Z) and it is easy to see that the convergence of

∑

k∈Z

ckψk is equivalent

to the convergence in norm of the symmetric partial sums. If we consider the ω-linear independence (the case where K = C

ω), this also holds due to the followingproposition. We skip the proof since it is elementary.

Proposition 1 Let ψ ∈ L2(R). Then Bψ is ω-linearly independent if and only if Bψis c0-linearly independent.

Periodization function pψ defined by (1) is obviously 1-periodic, so it can be consid-ered as a function on torus T := R/Z. Since ψ ∈ L2(R), it follows that pψ ∈ L1(T).It can be shown (see [6] for details and proof) that 〈ψ〉 is isometrically isomorphic tothe space L2(T, pψ) of square integrable functions with measure pψ(ξ)dξ . The roleof ψk’s is played by the family of exponentials {ek(ξ) = e2π ikξ : k ∈ Z}. Therefore,instead of considering the �p-linear independence of translates, we can consider the�p-linear independence of the system of exponentials in L2(T, pψ).

3 The Construction of Counterexamples

Since �2-linear independence of Bψ is equivalent to pψ > 0 a.e., this condition is alsothe necessary condition for �p-linear independence, for all p > 2. However, we show

J Fourier Anal Appl

that it is not a sufficient condition. First, we show that convergence of∑

k∈Z

ckek to 0

in L2(T, pψ) cannot be obtained if restricting on a set of Fourier coefficients of anintegrable function.

Proposition 2 Suppose that pψ > 0 a.e. and that there exists a nontrivial sequence(ck) such that

∑

k∈Z

ckek = 0

in L2(T, pψ). Then, there exists no f ∈ L1(T), f �= 0 such that ck = f (k), for allk ∈ Z.

Proof Suppose to the contrary, that (ck)k∈Z is a sequence of Fourier coefficients ofsome 0 �= f ∈ L1(T). We, then, have

∫

T

|Sn f (ξ)|2 pψ(ξ)dξ → 0, as n → ∞,

where Sn f = ∑|k|�n f (k)e2π ikξ . But then, the sequence of Cesàro sums {σn f } also

converges to 0 in L2(T, pψ) and, hence, there is a subsequence {σn j f√

pψ } whichconverges to 0 a.e. Due to the assumption, pψ is positive a.e., so it follows that

limj→∞(σn j f )(ξ) = 0, a.e.

On the other hand, using Fejér–Lebesgue’s theorem, one gets σn f (ξ) → f (ξ) a.e.,so we can conclude that ck = 0, for all k ∈ Z, which is in contradiction with theassumption that f is nontrivial. ��Remark 1 If we examine the proof above one more time, we see that it follows from

∑

k∈Z

ckek = 0 in L2(T, pψ)

that there exists a subsequence of partial sums {sn j } which converges to zero a.e.However, in general, we cannot conclude that ck = 0 for all k ∈ Z.

The preceding remark suggested that our problem could have been somehow relatedto the problem of uniqueness of trigonometric series. We recall the main definitionsand facts.

Definition 2 We say that E ⊆ T is a set of uniqueness if

∑

k∈Z

cke2π ikξ = 0, ξ /∈ E ⇒ ck = 0,∀k ∈ Z.

Otherwise, we say that E is a set of multiplicity.

J Fourier Anal Appl

It is not hard to see that any set of a positive Lebesgue measure is a set of multi-plicity. Menshov was the first to construct a set of Lebesgue measure zero which is aset of multiplicity. In other words, he showed that there existed trigonometric series∑

k∈Zcke2π ikξ converging to zero a.e. Such series are usually called the trigonometric

null-series. The structure of sets of uniqueness, as well as the properties of trigono-metric null-series have been widely investigated (see e.g. [9] or [11] for many resultsand references). However, the results concerning trigonometric null-series are mostlythe “negative” ones, which also reflects to our problem.

Sets of multiplicity can be characterized as in the following proposition.

Proposition 3 A compact set K ⊆ T is a set of multiplicity if and only if there existsa (Schwartz) distribution S such that

(1) |S(n)| → 0, as |n| → ∞(2) suppS ⊆ K .

Distributions satisfying condition (1) in Proposition 3 are also called pseudofunc-tions, whilst pseudomeasures are distributions having bounded Fourier coefficients.It is known that the Hausdorff dimension of a set of multiplicity can be zero, but forp > 2, a compact set having a Hausdorff dimension less than 2

p cannot support apseudomeasure with Fourier coefficients belonging to �p (see [8]).

Piatetski-Shapiro showed in 1954 that the term “distribution” in Proposition 3 can-not be replaced by “measure” having the same properties, although it is a more com-mon way used in construction of trigonometric null-series. However, in this section,we shall need the results concerning partial sums of such series and not the propertiesof the sets.

For a trigonometric series∑

k∈Zcke2π ikξ let

S∗(ξ) = supn

|sn(ξ)| (3)

denote the majorants of its symmetric partial sums. Observe that if such series is atrigonometric null series, then S∗ < ∞ a.e. and for a.e. ξ only finitely many |sn(ξ)|are bigger then 1. Observe also that S∗ cannot be bounded uniformly on ξ , due to theorthogonality of exponentials.

Proposition 4 Let 2 < p � ∞ and ψ ∈ L2(R). If there exists a function g ∈ L1(T)

and a trigonometric null-series∑

k∈Zcke2π ikξ such that (ck) ∈ �p and

pψ � g

(S∗)2a.e., (4)

then Bψ is �p-linearly dependent.

J Fourier Anal Appl

Proof If (ck) are coefficients of such series, then due to (4), it follows from Lebesgue’sdominated convergence theorem that

limn→∞ ‖

∑

|k|�n

c−kψk‖22 = lim

n→∞ ‖∑

|k|�n

ckek‖2L2(T,pψ)

= limn→∞

∫

T

|sn(ξ)|2 pψ(ξ)dξ = 0.

��One of the main problems in the theory of uniqueness was the problem of how

rapidly the coefficients of a trigonometric null-series can converge to zero, precisely,given a sequence (αn) of positive numbers such that

αn → 0,∑

n

α2n = ∞ (5)

does there exist a trigonometric null-series with coefficients dominated by (αn), i.e.

|cn| � αn, ∀n. (6)

The problem was studied in the 50’s by Littlewood, Wiener and Wintner, Schaeffer,Salem, Ivashev-Musatov and others, however, the problem had stayed unsolved untilthe positive answer was given in 1985 in the works of Pogosyan, [12], Arutyunyan,[1] and Körner, [10] (for discussion and further references see [11]).

Corollary 1 There exists ψ ∈ L2(R) such that pψ > 0 a.e. and Bψ is �p-linearlydependent for all p > 2.

Proof Let αn = 1√n

. Then, there exists a trigonometric null-series with coefficients

bounded by (αn) (see the discussion preceding the theorem). Therefore, it follows that(cn) ∈ �p, for all p > 2. Define

pψ(ξ) = g(ξ)

max{1, supn |sn(ξ)|2} ,

where g ∈ L1(T) and g > 0 a.e. Then pψ > 0 a.e., pψ ∈ L1(T) and it follows fromProposition 4. that Bψ is �p-linearly dependent for all p > 2. ��

Corollary 1 shows that condition pψ > 0 a.e. is not the sufficient condition for�p-linear independence, for any p > 2. Since we know that 1

pψ∈ L1(T) implies

minimality of the system (see [6]), it is also a sufficient condition for ω-linear inde-pendence. Hence, one could consider the conditions

1

pψ∈ Lr , for some r < 1,

1

pψ∈

⋂

r<1

Lr (7)

J Fourier Anal Appl

as the possible candidates for a sufficient condition. As we have seen that convergencein L2(T, pψ) cannot be obtained considering only the Fourier series of an integrablefunction, in the sequel we shall need the conditions which guarantee that a giventrigonometric series is a Fourier series. One of those conditions is the necessary andsufficient condition expressed in terms of Cesàro means (see e.g. [2] or [18]):

∫

T

|σn(ξ)− σm(ξ)|dξ → 0, as n,m → ∞. (8)

Simple use of Hölder’s inequality combined with the use of this fact gives an alternativeproof of the known fact that integrability of 1

pψimplies ω-linear independence. One

also sees that condition (8) implies that no trigonometric null-series can be found suchthat its partial sums converge to 0 in L1-norm. However, such sums can be boundedin L1-norm, and even positive, as it was proved by Weiss in [17] and, respectively, byKatznelson in [7]. The use of Egorov’s theorem and Hölder’s inequality proves that, ifa trigonometric null-series has positive partial sums, then it must satisfy the conditionlimn→∞ ‖sn‖r = 0, for all r < 1.

It has been shown in the work of Gevorkyan (see [5]) that the majorants of trigono-metric null-series cannot satisfy the condition

lim infλ→∞ λμ(ξ : S∗(ξ) � λ) = 0. (9)

On the other hand, Galstyan and Ovsepyan constructed in [4] trigonometric null-series that have positive partial sums, with coefficients tending to zero as rapidlyas possible (in a sense of (5) and (6)). Obtaining a situation of the subsequence ofpartial sums being very close to a martingale, after the main theorem they have made aremark that a modification of Doob’s inequality can be used to conclude that majorantsof partial sums satisfy the condition

supλ>0

λμ(ξ : S∗(ξ) � λ) < +∞. (10)

We state here the main result given in [4].

Theorem 1 (Galstyan, Ovsepyan)For any sequence (αn)n�0 which satisfies the con-dition

αn ↓ 0,∑

α2n = ∞, (11)

there exists a trigonometric null-series

a0 +∞∑

n=1

(an cos nξ + bn sin nξ) (12)

which has partial sums positive everywhere and

|a0| � α0, max(|an|, |bn|) � α2n ∀n � 1. (13)

J Fourier Anal Appl

The properties we need are summarized in the following lemma (note that some ofthem are, however, redundant).

Lemma 1 There exists a trigonometric null-series∑

k∈Zcke2π ikξ such that

(a) supn ‖sn‖1 < ∞,(b) limn→∞ ‖sn‖r = 0, ∀r < 1,(c) S∗ ∈ L1,∞, i.e. λμ(ξ : S∗(ξ) � λ) � C < ∞, ∀λ > 0(d) (ck) ∈ �p, ∀p > 2,

Proof The lemma follows using Theorem 1 (i.e. Theorem 2 in [4]) and the commentsafter the theorem (page 13, relation (48)). ��Corollary 2 For any ε > 0 there exists ψ ∈ L2(R) such that 1

pψ∈ L1−ε(T) and Bψ

is �p-linearly dependent for all p > 2.

Proof Let (ck) be the coefficients of a series with the properties as in Lemma 1 andlet ε > 0 be arbitrary. Take pψ as in Corollary 1, with g defined as

g(ξ) = max{1, (S∗(ξ))1−ε}.

Since S∗ ∈ L1,∞, we have S∗ ∈ Lr , for each r < 1. Hence, g ∈ L1(T) and, usingTheorem 4, it follows that Bψ is �p-linearly dependent, for all p > 2. Moreover,

1

pψ(ξ) = max{1, S∗(ξ)1+ε}

and, since we have ((S∗)1+ε)1−ε = (S∗)1−ε2 ∈ L1(T), the result follows. ��Theorem 2 There exists ψ ∈ L2(R) such that 1

pψ∈ ⋂

r<1 Lr (T) and Bψ is �p-linearly dependent for all p > 2.

Proof Let (ck) be the coefficients of a series from Lemma 1. We can assume thatC > 1. Define

Am :={ξ ∈ T : C

am−1� S∗(ξ) < C

am

}, m � 2

and

A1 :={ξ ∈ T : S∗(ξ) � C

a1

},

where (am) is a decreasing sequence of positive numbers such that limm→∞ am = 0,

and∞∑

m=1

am−11m < ∞ (take a1 = 1). Observe that Am’s are mutually disjoint, T =

⋃∞m=1 Am and

μ(Am) � μ(ξ : S∗(ξ) � C

am−1) � am−1, m � 2.

J Fourier Anal Appl

Define

pψ(ξ) = 1

χA1(ξ)+∞∑

m=2(S∗(ξ))1+ 1

m χAm (ξ)

. (14)

Let ε > 0 be arbitrary. Take M ∈ N such that

∞∑

m=M+1

(am−1)1m <

ε

3 max{1, supn ‖sn‖1} .

Since |sn|2χA1 � 1, for all n and ξ , due to Lebesgue’s dominated convergence theorem,we can choose n1 ∈ N such that

∫

T

|sn(ξ)|2χA1(ξ)dξ <ε

3,

for all n � n1. Since ‖sn‖r → 0 as n → ∞, for all r < 1, we can choose n2 ∈ N

such that

M∑

m=2

∫

T

|sn(ξ)|1− 1m dξ <

ε

3

for all n � n2.It follows that, for sufficiently large n, one has

∫

T

|sn(ξ)|2 pψ(ξ)dξ =∫

T

|sn(ξ)|2χA1(ξ)dξ

+M∑

m=2

∫

T

|sn(ξ)|2 1

(S∗)1+ 1m (ξ)

χAm (ξ)dξ

+∞∑

m=M+1

∫

T

|sn(ξ)|2 1

(S∗)1+ 1m (ξ)

χAm (ξ)dξ

� ε

3+

M∑

m=2

∫

T

|sn(ξ)|1− 1m dξ +

∞∑

m=M+1

∫

T

|sn(ξ)|1− 1m χAm (ξ)dξ

� 2ε

3+

∞∑

m=M+1

( ∫

T

(|sn(ξ)|1− 1

m

) mm−1

)m−1m

(∫

T

(χAm (ξ))mdξ

) 1m

= 2ε

3+

∞∑

m=M+1

(‖sn‖1)m−1

m (μ(Am))1m < ε,

J Fourier Anal Appl

with Hölder’s inequality applied to the functions f = |sn|1− 1m ∈ L

mm−1 and g =

χAm ∈ Lm . Hence, we have∑

k∈Zcke2π ikξ = 0 in L2(T, pψ).

We shall prove first that 1pψ

∈ ⋂k∈N

L1− 1k .

Let k ∈ N be arbitrary and put r = 1 − 1k . We, then, have

∫

T

( 1

pψ(ξ)

)rdξ =

∫

T

χA1(ξ)dξ +∞∑

m=2

∫

T

((S∗(ξ))1+ 1m )

rχAm (ξ)dξ

� 1 +k−1∑

m=2

∫

Am

(S∗(ξ))(1+ 1m )r dξ +

∞∑

m=k

∫

T

(S∗(ξ))(1+ 1m )rχAm (ξ)dξ

� 1 +k−1∑

m=2

( C

am

)(1+ 1m )r

am−1 +∞∑

m=k

∫

T

(S∗(ξ))(1+ 1k )(1− 1

k )χAm (ξ)dξ

� 1 + Kr +∫

∞⋃m=k

Am

(S∗(ξ))1− 1k2 dξ

� 1 + Kr + ‖S∗‖1− 1k2< +∞,

so 1pψ

∈ L1− 1k . For r < 1, choose k ∈ N such that r < 1 − 1

k . Hence 1pψ

∈ Lr andthe theorem follows. ��

We have, thus, proved that neither condition in (7) is a sufficient condition for�p-linear independence.

Remark 2 We cannot improve the result of the previous theorem in a sense that1

pψ∈ L1,∞. This is a consequence of the mentioned result of Gevorkyan concerning

majorants of trigonometric null-series. Suppose to the contrary, that pψ is defined asin the preceding theorem and that there exists K > 0 such that λμ( 1

pψ� λ) � K ,

∀λ > 0. Since we know (by (9)) that

lim infλ→∞ λμ(S∗(ξ) � λ) > 0,

it follows that for every subsequence (λm) there exists ε > 0 such that for any suffi-ciently large m we have

λmμ(S∗(ξ) � λm) � ε.

Consider λk = Cak

. Observe that if m � k + 1 and ξ ∈ Am , then we have S∗(ξ) � Cak

,a.e., so it follows that

μ((S∗(ξ))1+ 1m χAm (ξ) � C

ak) = μ(Am), ∀m � k + 1.

J Fourier Anal Appl

On the other hand, if m � k − 1, we can conclude that

μ((S∗(ξ))1+ 1

m χAm (ξ) � C

ak

)= 0.

Indeed, if ξ ∈ Ak−1, we have

(S∗(ξ))1+ 1k−1 �

( C

ak−1

)1+ 1k−1.

If we had(

Cak−1

)1+ 1k−1 � C

ak, it would mean that

(C

ak−1

) 1k−1 �

(Cak

) 1k, but 1

(ak)1/k →

∞, due to the assumption in the proof of the Theorem 2.Therefore, if k is sufficiently large, one gets

K � λkμ(1

pψ� λk)

�∞∑

m=2

C

akμ((S∗(ξ))1+ 1

m χAm (ξ) � C

ak)

� C

akμ

(S∗(ξ) �

( C

ak

)1− 1k+1

)

> ε · C

ak· (ak)

1− 1k+1

C1− 1k+1

= ε · C1

k+1

(ak)1

k+1

.

But then we have supk1

(ak)1

k+1< +∞, which derives a contradiction.

4 A New Sufficient Condition

In order to have a notion that matches the �p-linear independence, we slightly changethe Definition 2.

Definition 3 Let 2 < p � ∞. We say that E ⊆ T is a set of p-uniqueness (shortly,Up-set) if

(ck) ∈ �p,∑

k∈Z

cke2π ikξ = 0, ξ /∈ E ⇒ ck = 0,∀k ∈ Z.

Otherwise, we say that E is a set of p-multiplicity (shortly, Mp-set).

Observe that, due to Cantor–Lebesgue’s theorem, M∞-sets are exactly the sets fromDefinition 2. Although the notions share a very similar name, observe that the definition

J Fourier Anal Appl

of �p-set of uniqueness from [15] is of a different nature, since the interesting examplesof the �p sets of uniqueness are found amongst the sets of the positive (Lebesgue)measure and there do not exist such sets for the case p > 2, whilst there do not existany nontrivial sets of p-multiplicity of Lebesgue measure zero for any p � 2.

In the sequel, we shall often need the condition

∞∑

l=0

∑

|k|�l

| f (k)| < ∞ (15)

in order to use the Rajchman’s multiplication theory (see e.g. [18], Chapter IX, 4).Namely, it is known that if f is a continuous function satisfying (15) and if cn → 0,as n → ∞, then

∑

|n|�N

Cne2π inξ − f (ξ)∑

|n|�N

cne2π inξ → 0,

as N → ∞, uniformly on ξ , where Cn is defined as Cn = ∑k ck f (n − k). Condition

(15) is satisfied e.g. if f (n) = O( 1|n|3 ).

Theorem 3 Let 2 < p � ∞ and ψ ∈ L2(R). If there exists a collection of openintervals {I j ; j ∈ J } such that 1

pψχI j ∈ L1(T) for all j ∈ J and

⋃j∈J I j = T \ C

where C is a Up-set, then Bψ is �p-linearly independent.

Proof Suppose to the contrary, that there exists a nontrivial sequence (ck) ∈ �p suchthat

limn→∞

∫

T

|sn(ξ)|2 pψ(ξ)dξ = 0,

Since for all j ∈ J we have 1pψχI j ∈ L1(T), using Hölder’s inequality, one gets

( ∫

I j

|sn(ξ)|dξ)2(

∫

I j

1

pψ(ξ)dξ

)−1�

∫

I j

|sn(ξ)|2 pψ(ξ)dξ −→ 0,

as n → ∞, i.e.

limn→∞

∫

I j

|sn(ξ)|dξ = 0.

Let now g be a continuous function satisfying (15), that vanishes outside I j . We canchoose g such that g ∈ C∞(T), g > 0 everywhere on I j and g = 0 on I c

j . Define

J Fourier Anal Appl

(Cn) as a convolution of sequences (ck) and (g(k)), i.e.

Cn =∞∑

k=−∞ck g(n − k).

Using Rajchman’s theorem, it follows that

limn→∞ |

∑

|k|�n

Cke2π ikξ − g(ξ)∑

|k|�n

cke2π ikξ | = 0,

uniformly on ξ . Hence,

∫

T

|∑

|k|�n

Cke2π ikξ |dξ �∫

T

|∑

|k|�n

Cke2π ikξ − g(ξ)∑

|k|�n

cke2π ikξ |dξ

+∫

T

|g(ξ)||∑

|k|�n

cke2π ikξ |dξ −→ 0,

as n → ∞. Therefore, it follows that∑

k∈Z

Cke2π ikξ converges to zero in L1(T). We

conclude, using (8), that it is a Fourier series. But, then the Fejér–Lebesgue’s theoremyields Ck = 0, for all k ∈ Z. Hence, we have

∑

|k|�n

Cke2π ikξ = 0, for all n, for all ξ ∈ T.

and

limn→∞ g(ξ)

∑

|k|�n

cke2π ikξ = 0,

uniformly on ξ . Since, g(ξ) > 0 for all ξ ∈ I j , it follows that

limn→∞

∑

|k|�n

cke2π ikξ = 0, for all ξ ∈ I j .

Finally, we conclude that

∑

k∈Z

cke2π ikξ = 0, for all ξ ∈⋃

j∈J

I j = T \ C.

But, since C is a Up-set, it follows that

ck = 0, ∀k ∈ Z.

��

J Fourier Anal Appl

For g ∈ C(T), denote by Zg the set of zeroes of g, i.e. the set {ξ ∈ T : g(ξ) = 0}.Obviously, a complement of such set is an open set which can be expressed as a unionof intervals on which 1

pψis bounded. Observe also that if Bψ is �p-linearly independent

and pψ ′ � pψ a.e., then Bψ ′ is �p-linearly independent. Hence, we have the followingcorollary.

Corollary 3 Let 2 < p � ∞ and ψ ∈ L2(R). If there exists g ∈ C(T) such thatg � pψ a.e. and Zg is a Up-set, then Bψ is �p-linearly independent.

The preceding theorems give an answer to some simple cases, e.g. when pψ(ξ) = ξ ,that we did not know before (since 1

pψ/∈ L1(T) we did not know whether the system

was �p-linearly independent or not). However, we can now give an answer to somemore interesting examples.

Namely, when Menshov gave an example of a set of multiplicity having a Lebesguemeasure zero, he actually constructed a set of a Cantor type, i.e. a perfect, nowheredense set constructed in a way as the “standard” Cantor set. However, such sets canalso be sets of uniqueness, as it is the case with the standard Cantor set. The mostfamous theorems that characterize Cantor type sets of uniqueness are the theoremsof Salem and Zygmund (more details can be found in [9]) where the characterizationis given in terms of Pisot numbers. Since it is known that any closed set can be thezero set of a continuous function, we get a variety of examples. Our next task was tocheck if there existed at all, a continuous function with its zero set being an Mp-setand system Bψ being �p-linearly dependent. The answer appeared to be positive.

Theorem 4 Let 2 < p � ∞. There exists ψ ∈ L2(R) such that there is a collectionof intervals {I j : j ∈ J }, where T \ ⋃

j∈JI j is an Mp set,

1

pψχI j ∈ L∞(T), ∀ j ∈ J

and Bψ is �p-linearly dependent. Moreover, for any closed set M ⊆ T of a Cantor-type, having Lebesgue measure zero, which is an Mp-set, there exists pψ ∈ C(T) suchthat Zpψ = M and Bψ is �p-linearly dependent.

Proof Let M be a closed set of a Cantor-type, of Lebesgue measure zero which is anMp-set. We, then, have

T \ M =⋃

m

Im,

where Im = ⋃j J j denotes the union of open intervals removed in the m-th step of

the construction. Let∑

k∈Z

cke2π ikξ be a trigonometric series which converges to zero

outside M . For any interval J j ⊆ Im ⊆ T \ M we define (Cn) as

Cn =∞∑

k=−∞ck g j (n − k),

J Fourier Anal Appl

where g j ∈ C∞(T) is taken to be positive everywhere on J j , and vanish on J cj . Due

to Rajchman’s theorem, we have

∑

|n|�N

Cne2π inξ − g j (ξ)∑

|n|�N

cne2π inξ −→ 0,

as N → ∞, uniformly on ξ . Since g j (ξ) = 0 everywhere on J cj and

∑k∈Z

cke2π ikξ =0, for all ξ ∈ J j , it follows that

∑

|n|�N

Cne2π inξ −→ 0,

as N → ∞ for all ξ ∈ T. It, thus, follows that Cn = 0, for every n ∈ Z. Furthermore,it means that

g j (ξ)∑

|k|�n

cke2π ikξ −→ 0,

as n → ∞, uniformly on ξ , so there exists K j < ∞ such that

supn

|sn(ξ)| � K j

g j (ξ),

for all ξ ∈ J j . Define

pψ(ξ) = 1

max{1, supn |sn(ξ)|2} .

Since we have

supn

|sn(ξ)|2 �K 2

j

g2j (ξ)

, ξ ∈ T,

and 1g2

jis continuous and positive everywhere on J j , there is a collection of subintervals

of J j on which 1g2

jand, hence, supn |sn(ξ)|2 is bounded and whose union is equal to

J j .

Moreover, we can take√

pψ(ξ) = g j (ξ)

K j, for ξ ∈ J j and pψ(ξ) = 0, for ξ ∈ M . If

we assume that K j > 1 for all j and g j are sufficiently good, one may conclude thatpψ ∈ C(T), Zpψ = M and

∫

T

|sn(ξ)|2 pψ(ξ)dξ → 0,

as n → ∞, using Lebesgue’s dominated convergence theorem. ��

J Fourier Anal Appl

5 Characterization Theorem (A Partial Answer)

Finding a condition equivalent to �p-linear independence seems to be a difficult prob-lem since, obviously, the periodization function must depend on the properties ofpartial sums of a series

∑k∈Z

cke2π ikξ . Observe that Proposition 4 gives a sufficientcondition for a system to be �p-linearly dependent. However, it appeared to be moreuseful for constructing counterexamples (starting with the sequence (ck)) than forfinding coefficients (ck) when given a function pψ . Indeed, given a function pψ , itis difficult to give an answer whether 1√

pψdominates pointwisely the majorants of

a trigonometric null-series. We have showed that such condition is even a necessarycondition when

√pψ satisfies condition (15), but the proof is similar to the proof of

Theorem 5. Nevertheless, if we put some additional assumptions on Bψ we can givea characterization of �p-linear independence in terms of Up-sets.

If g ∈ L1(T), denote by Zg the set of zeroes of the continuous version of g, if suchfunction exists (i.e. Zg := Zh , where h ∈ C(T) and g = h a.e.). Observe that, if gsatisfies (15), then there exists such representative.

Theorem 5 Let 2 < p � ∞ and ψ ∈ L2(R). Suppose that there exists g ∈ L1(T)

having the following properties:

(a)√

g satisfies (15),(b) g

pψ∈ L1(T),

pψg ∈ L1(T).

Then Bψ is �p-linearly independent if and only if Zg is a Up set.

Proof Suppose that Zg is a Up-set. We want to show that Bψ is �p-linearly indepen-dent. Suppose to the contrary, that there exists a nontrivial (ck) ∈ �p such that

‖∑

|k|�n

ckek‖L2(T,pψ) → 0, as n → ∞.

Then we also have

‖√g∑

|k|�n

ckek‖L2(T,pψg )

→ 0, as n → ∞.

Let the sequence (Cn) be defined as

Cn =∞∑

k=−∞ck

√g(n − k).

Since the function√

g satisfies (15), the use of the Rajchman’s multiplication theoryyields

‖∑

|k|�n

Ckek‖L2(T,pψg )

� ‖∑

|k|�n

Ckek − √g

∑

|k|�n

ckek‖L2(T,pψg )

+‖√g∑

|k|�n

ckek‖L2(T,pψg )

−→ 0,

J Fourier Anal Appl

as n → ∞. However, since 1pψ

g ∈ L1(T), it follows that

Ck = 0, for all k ∈ Z.

Using Rajchman’s multiplication theorem one more time, one concludes that

√h(ξ)

∑

|k|�n

cke2π ikξ → 0,

as n → ∞, for all ξ ∈ T, where h ∈ C(T), h = g a.e. Since h is positive everywhereon Zc

g , it follows that

limn→∞

∑

|k|�n

cke2π ikξ = 0, ξ /∈ Zg.

But, we assumed that Zg was a Up-set. Therefore, it follows that

ck = 0, for all k ∈ Z.

Suppose now that Zg is an Mp-set. Then, there exists a nontrivial sequence (ck) ∈ �p

such that

∑

k∈Z

cke2π ikξ = 0, ξ /∈ Zg.

If we put

Cn =∞∑

k=−∞ck

√g(n − k),

then, using Rajchman’s multiplication theorem, we get

∑

|k|�n

Cke2π ikξ → 0,

as n → ∞, for each ξ ∈ T, since h(ξ)∑

|k|�n cke2π ikξ → 0 for all ξ ∈ T (h isagain the continuous representative). But then, it follows that Ck = 0, for all k ∈ Z.Therefore, we conclude that

√g(ξ)

∑

|k|�n

cke2π ikξ → 0,

J Fourier Anal Appl

as n → ∞, uniformly a.e. on ξ . Since 1g pψ ∈ L1(T), it follows from Lebesgue’s

dominated convergence theorem that

‖√g∑

|k|�n

ckek‖L2(T,pψg )

→ 0,

as n → ∞. Hence,

limn→∞ ‖

∑

|k|�n

ckek‖L2(T,pψ) = 0,

i.e. Bψ is �p-linearly dependent. ��In general, it is not even clear if Up-sets would play part in a characterization

theorem. Indeed, it makes no sense to discuss whether the corresponding zero set is aset of uniqueness, the periodization function pψ being positive a.e. and defined up toa set of Lebesgue measure zero. Thus, the general problem of �p-linear independencestill remains unsolved.

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