Wiener's `closure of translates’problem and Piatetski-Shapiro's uniqueness phenomenon

Abstract

N. Wiener characterized the cyclic vectors (with respect to translations) in ℓ p (Z) and L p (R), p = 1, 2, in terms of the zero set of the Fourier transform. He conjectured that a similar characterization should be true for 1 < p < 2. Our main result contradicts this conjecture.is non-zero almost everywhere.(ii) c is cyclic in ℓ 1 (Z) if and only if c(t) has no zeros.Part (i) is a consequence of the unitarity of the Fourier transform. Part (ii) is more delicate, the proof is based on the fact that the space ℓ 1 (Z) is a convolution algebra.In both cases the result can be stated as follows: c is a cyclic vector if and only if the set Z c := {t ∈ T : c(t) = 0} of the zeros of the Fourier transform (1) is "small" in a certain sense. Wiener conjectured (see [24], p. 93) that a similar result should be true for ℓ p spaces, at

Cyclicity in ℓ spaces and zero sets of the Fourier transforms

“…Wiener asked if the same phenomenon holds for u ∈ ℓ p (Z), 1 < p < 2. Lev and Olevskii, in [14], answered this question negatively. We cannot characterize the bicyclicity of u ∈ ℓ 1 (Z) only in terms of the zero set of u: for 1 < p < 2 there exist u, v ∈ ℓ 1 (Z) such that Z( u) = Z( v) and one is bicyclic in ℓ p (Z) and the other is not.…”

“…is dense in ℓ p (Z) and u is called bicyclic in ℓ p (Z) if the linear span of {(u n−k ) n∈Z , k ∈ Z} is dense in ℓ p (Z). Note that the bicyclicity in this paper is defined as cyclicity in [13] and [14].…”

Discrete translates in Lp(R)

“…Denote by F the inverse Fourier transform of f , and by ψ the Fourier transform of Ψ. From (7), for every n ∈ Z we obtain:…”

“…However, this condition is not necessary. Moreover, the spanning property of the translates of cannot be characterized in terms of the zero set of , see .…”

Singular distributions, dimension of support, and symmetry of Fourier transform

“…The abbreviations are KO for [11], Ri for the Riesz analyticity theorem, PS for Piatetski-Shapiro [18] and Ra for Rajchman [19]. The arrow marked LO comes from [15] and noting that the set constructed in [15] is a Helson set, so it cannot support even a measure µ with µ(n) → 0. The unmarked arrow on the top-left is trivial, while the unmarked arrow on the bottom-right and the two unmarked arrows on the bottom left follow from the other arrows in their diagrams.…”