Distribution of zero subsequences for Bernstein space and criteria of completeness for exponential system on a segment

TL;DR

This paper characterizes the zero subsequences for Bernstein spaces and establishes criteria for the completeness of exponential systems on segments, advancing understanding of entire functions and exponential bases.

Contribution

It provides a complete description of non-uniqueness sequences in Bernstein spaces and criteria for exponential system completeness on segments.

Findings

Characterization of zero subsequences for Bernstein spaces.

Criteria for exponential system completeness in function spaces.

Conditions for non-uniqueness sequences in entire functions.

Abstract

For $\sigma\in (0,+\infty)$, denote by $B_\sigma^{\infty}$ the Bernstein space (of type $\sigma$) of all entire functions of exponential type $\leq \sigma$ bounded on real axis $\R$. Let $I_d\subset \R$ be a segment of length $d>0$. We announce complete description of non-uniqueness sequences of points for $B_\sigma^\infty$ and criteria of completeness of exponential system in $C(I_d)$ or $L^p(I_d)$ to within one or two exponential functions.