Piatetski-Shapiro's phenomenon and related problems

TL;DR

This thesis explores advanced Fourier analysis topics, extending Piatetski-Shapiro's phenomenon to new spaces and revealing limitations in characterizing functions by their Fourier zeros.

Contribution

It extends Piatetski-Shapiro's phenomenon to  spaces and shows that functions in  spaces cannot be characterized solely by their Fourier transform zeros.

Findings

Extended Piatetski-Shapiro's phenomenon to  spaces

Demonstrated limitations in characterizing functions via Fourier zeros for 1<p<2

Contrasted results with classical Wiener theorems

Abstract

This Ph.D. thesis, prepared under the supervision of Prof. Alexander Olevskii, is concerned with some problems in two areas of Fourier Analysis: uniqueness theory of trigonometric expansions, and the theory of translation invariant subspaces in function spaces. Our main result in the first area extends to $\ell_q$ spaces ($q > 2$) a deep phenomenon found by Piatetski-Shapiro in 1954 for the space $c_0$. The approach we developed also enabled us to get a result in the second mentioned area, which a priori does not look connected with the first one. The result (maybe, a bit surprising) is: one cannot characterize the functions in $\ell_p(\Z)$ or $L^p(\R)$, $1 < p < 2$, whose translates span the whole space, by the zero set of their Fourier transform. This should be contrasted against the classical Wiener theorems related to the cases $p=1,2$.