Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces

TL;DR

This paper investigates the existence of unconditional Schauder frames of exponentials in $L^p$ spaces, showing that while bases are impossible, Schauder frames can exist under certain conditions, but not in spaces with nonempty interior.

Contribution

It demonstrates that the non-existence of unconditional bases of exponentials in $L^p$ spaces does not extend to Schauder frames, and establishes conditions under which such frames cannot exist.

Findings

Unconditional bases of exponentials do not exist in $L^p( ext{finite measure})$ spaces for $p e 2$.

Unconditional Schauder frames of exponentials can exist in these spaces.

No unconditional Schauder frames of exponentials exist in $L^p( ext{space with nonempty interior})$ for $p e 2$.

Abstract

It is known that there is no unconditional basis of exponentials in the space $L^p(\Omega)$, $p \ne 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(\Omega)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $\Omega$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $p \ne 2$.