A simple observation about compactness and fast decay of Fourier coefficients

TL;DR

This paper proves a general result about the decay of Fourier and frame coefficients in Banach spaces, linking compact embeddings and weakly null sequences, with applications to $L^p$ spaces and Hilbert frames.

Contribution

It introduces a new general theorem connecting compact embeddings and coefficient decay, applied to Fourier and frame analysis in Banach and Hilbert spaces.

Findings

Existence of a sequence tending to zero controlling coefficient decay

Application to Fourier coefficients in $L^p(\

Abstract

Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that $|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every $y\in Y$. We prove this result and we use it for the study of fast decay of Fourier coefficients in $L^p(\mathbb{T})$ and frame coefficients in the Hilbert setting.