Student Seminar Talk TRANSLATION-INVARIANT SUBSPACES IN l1 AND RANDOM FOURIER SERIES

We are interested in the following problem: to describe the closed translation-invariant subspaces of lp(Z). For l2 there is a complete solution due to Wiener, in terms of zeros of Fourier transform. However, the situation when p 6= 2 is much harder. In this talk we will present a proof of Malliavin’s theorem which shows the difficulty of characterizing all invariant subspaces of l1. We will use a probabilistic approach due to Kahane. 1. Translation invariant subspaces We work in the space l2(Z). Consider the translation (or shift) operator S : {c(n)} 7→ {c(n− 1)}, which is a unitary operator in l2(Z). A closed subspace E is a traslation-invariant subspace if S(E) = E (that is, it is invariant to both right and left shift). One of the motivations for working with traslation-invariant subspaces may come from the theory of representation and approximation of signals. We are interested in the following problem: to describe the closed translation-invariant subspaces in l2(Z). The solution of this problem is due to Wiener. To each {c(n)} ∈ l2 there corresponds it’s Fourier transform (1.1) f(t) = ∑ n∈Z c(n)e (t ∈ T), which is a function in L(T) defined up to a set of measure zero. For a (measurable) set Λ ⊂ T we denote E(Λ) = {c ∈ l2(Z) : f(t) = 0 a.e. on Λ}. Because translation of the sequence {c(n)} corresponds to multiplication of f(t) by e, it is clear that E(Λ) is a (closed) translation-invariant subspace of l2(Z). It turns out that no other invariant subspaces exist, namely: Theorem 1.1 (Wiener). To each closed translation-invariant subspace E of l2(Z) there corresponds a unique (up to a set of measure zero) Λ such that E = E(Λ). The space E(Λ) can be interpreted as the space of discrete signals which contain no pure components {e} with frequencies t ∈ Λ. 2. Scwartz-Malliavin phenomenon We now consider the space l1(Z). In this case the function (1.1) is defined everywhere on T and is a continuous function, so its set of zeros is closed. For a given compact K ⊂ T, the role of E(Λ) is now played by (2.1) I(K) = {c ∈ l1 : f(t) = 0 ∀t ∈ K}. This is an expository talk delivered on December 18 and 25, 2005 in the Student Seminar, Tel-Aviv university. 1