$\ell^2$-Linear Independence for the System of Integer Translates of a Square Integrable Function

We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.


Introduction
Given a square integrable function ψ ∈ L 2 (R) , many properties of the system of integer translates can be completely described in terms of properties of the 1-periodic function p ψ (ξ) = k∈Z |ψ(ξ + k) | 2 , ξ ∈ R, called the periodization function of ψ (note that p ψ ∈ L 1 (T)). Systems of integer translates arise in the context of wavelet analysis and, more generally, in the theory of shift invariant spaces. We refer the reader to the work of Hernández,Šikić, Weiss, and Wilson, [1] and to references contained therein for a comprehensive summary of these properties.
In this paper we focus on ℓ 2 -linear independence. First of all, let us pass to recall different concepts of independence in great generality. Definition 1. Let (e n ) n∈N be a sequence in a Hilbert space H. We say that (i) (e n ) n∈N is linearly independent if every finite subsequence of (e n ) n∈N is linearly independent.
(ii) (e n ) n∈N is ℓ 2 -linearly independent if whenever the series +∞ n=0 c n e n is convergent and equal to zero for some coefficients (c n ) n∈N ∈ ℓ 2 (N), then necessarily c n = 0 for all n ∈ N.
c n e n is convergent and equal to zero for some scalar coefficients (c n ) n∈N , then necessarily c n = 0 for all n ∈ N.
(iv) (e n ) n∈N is minimal if for all k ∈ N, e k / ∈ span{e n , n = k}.
Hence by k∈Z c k T k ψ = 0 we mean lim n→+∞ |k|≤n c k T k ψ = 0. So B ψ is ℓ 2 -linearly independent if and only if whenever {c k } ∈ ℓ 2 and lim n→+∞ |k|≤n c k T k ψ 2 = 0, then necessarily c k = 0 for all k ∈ Z.
Relations between the various type of independence for B ψ and properties of the periodization function are summarized in the following scheme: In particular, it is known that A question raised in [1] is the following: Is the converse true? Sikić, and Speegle [5] have given a positive answer if p ψ is bounded; see also Paluszyński's paper [4], where it is proved that p ψ (ξ) > 0 a.e. is equivalent to L 2 -Cesàro linear independence of B ψ . The latter means that if the Cesàro averages tend to zero in L 2 norm, then necessarily c k = 0, for all k ∈ Z.
The approach we used in addressing this problem was global in nature: rather than examining the assumptions to be put on a single ψ, we preferred to analyze the issue as a whole, for all ψ ∈ L 2 (R) .
The result is that the converse is true for any ψ if and only if the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset A of [0, 1], with positive measure |A|, we can find a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.
By support of f ∈ L 1 (T) (denoted suppf ) we mean the smallest closed set S such that f (ξ) = 0 almost everywhere in the complement of S.
After the proof of the main result in Section 2, in Section 3 we discuss the existence of such a good function for any measurable set A ⊂ [0, 1]. As far as we know existence is obtained as a corollary of general results by Kislyakov and Vinogradov, although we realize that there may be other direct proofs that we are not aware of.
We end with some notations.
For f ∈ L 1 (R) the Fourier transform iŝ The author is grateful to Professor Guido Weiss for having introduced her to the subject.

Main result
In this section we prove the main result. The first step requires uniformly boundedness only in the complement of the set A.
By b) there exists 0 = f ∈ L 2 (T), such that both 1. and 2. hold. Now a simple calculation shows By a.e. convergence of the partial sums to f , and supp f ⊂ A, we get a.e.
Note that, so that A = {ξ ∈ [0, 1] : p ψ (ξ) = 0}. Hypothesis a) applied to ψ implies that B ψ is not ℓ 2 -linear independent, so there exists a non zero sequence {c k } ∈ ℓ 2 (Z) such that lim n |k|≤n There is a unique f ∈ L 2 (T) such thatf (k) = c −k . We shall show that the partial sums S n (f ) are uniformly bounded in A c . Once we have proved this, it is easy to show that f (ξ) = 0 a.e. in A c . Indeed So, for n → +∞, the left hand side tends to zero by (2), while, by uniform boundedness and the a.e. convergence of the partial sums, the right hand side tends to A c | f (ξ) | 2 dξ.
In order to prove the uniform boundedness of the partial sums, we shall apply the uniform boundedness principle to the following bounded linear operators T n : L 1 (T) → C, First, it is known that Furthermore, for g in the dense subspace L 1 (T) ∩ L 2 (T) of L 1 (T), we have by (2) So, see Hutson and Pym [2], there exists a unique T (g), for all g ∈ L 1 (T). In particular, for any fixed g ∈ L 1 (T), we have definitively which implies sup{| T n (g) | : n ∈ N} < +∞.
By uniform boundedness principle we get and everything is proved. Then S n (f ) converges uniformly in any closed subinterval of the intervals (a n , b n ) contiguous to A, but a priori nothing can be said for n (a n , b n ) = A c . Indeed, to prove b) implies c), it is sufficient to assume that A does not contain an interval, otherwise statement c) is true regardless of b).
We are going to show, by a density argument, that the same function f provided by b) works well.
So let f and M as in b). Let ξ 0 ∈ A and n ∈ N. Since S n (f ) is continuous in ξ 0 , we can find an open neighborhood I 0 of ξ 0 such that There exists at least one ξ ∈ A c ∩ I 0 (otherwise A contains an interval), and finally | S n (f )(ξ 0 ) |≤ 1+ | S n (f )(ξ) |< 1 + M.

Looking for a nice function
To the best of my knowledge, the existence of a nice function f satisfying condition c), for a given set A ⊂ [0, 1], can be derived by the following theorem in Kislyakov's paper [3,Theorem 4], whose proof relies also upon a result by Vinogradov [6]. The latter makes use of Carleson almost everywhere convergence theorem.
We recall first some basic notations: U ∞ denotes the space of functions f ∈ L ∞ (T) for which the following norm is finite f U ∞ = sup n≤k≤mf (k)ξ k , n, m ∈ Z, n ≤ m, ξ ∈ T .