Localization of trigonometric polynomials and the Lev-Tselishchev lemma

TL;DR

This paper investigates the limitations of the Lev-Tselishchev localization lemma in constructing quasi-bases from shifted functions in certain $L^p$ spaces, revealing it does not extend to smaller $p$ values.

Contribution

It demonstrates that the Lev-Tselishchev localization lemma fails to apply for $L^p$ spaces with small $p$, clarifying the boundaries of its applicability.

Findings

The lemma does not extend to $p o 1$.

Limitations are shown for constructing quasi-bases in small $p$ spaces.

The result refines understanding of localization in $L^p$ spaces.

Abstract

The paper shows that the localization lemma used by N. Lev and A. Tselishchev in solving the problem of constructing quasi-bases from uniformly separated shifts of a function in the space $L^p (\mathbb R) $ for $ p > ( 1 + \sqrt 5 )/2 $, is not carried over to small $p$.