A simple observation about compactness and fast decay of Fourier coefficients

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 $X^{\ast }$. Then there exists a sequence $\{{\epsilon }_n\}\searrow 0$ such that $|a_n(y)|\le {\epsilon }_n\Vert y{\Vert }_Y$ 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.

2 pages; submitted to a Journal