Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials

TL;DR

This paper investigates how the completeness and frame properties of certain exponential systems are affected by the gaps in the index set, linking these properties to the uniqueness sets of lacunary polynomials.

Contribution

It establishes a connection between the gaps in the index set of exponential systems and their completeness, revealing differences from classical systems and relating to lacunary polynomial zeros.

Findings

Gaps in the index set alter completeness and frame properties.

The phenomena are connected to uniqueness sets for lacunary polynomials.

Differences from classical exponential systems are characterized.

Abstract

Let $\Gamma$ be a subset of $\{0,1,2,...\}$. We show that if $\Gamma$ has `gaps' then the completeness and frame properties of the system $\{t^ke^{2\pi i nt}: n\in\mathbb{Z},k\in\Gamma\}$ differ from those of the classical exponential systems. This phenomenon is closely connected with the existence of certain uniqueness sets for lacunary polynomials.