A simple observation about compactness and fast decay of Fourier coefficients

Abstract

Let X be a Banach space and suppose YX is a Banach space compactly embedded into X, and (ak) is a weakly null sequence of functionals in X � . Then there exists a sequence fng & 0 such that jan(y)j � nkykY for every n 2 N and every y 2 Y. We prove this result and we use it for the study of fast decay of Fourier coefficients in L p (T) and frame coefficients in the Hilbert setting. 1. Motivation One of the classical problems in Harmonic Analysis is the study of the relationship that exists between decay properties of the Fourier coefficientscn(f) = 1 2� R 2� 0 f(t)e int dt of a 2�-periodic function f : T ! C and its membership to several function spaces. Just to mention a few well known examples, we show the following list of results: � Riemann-Lebesgue Lemma sates that for f 2 L 1 (T) the Fourier coefficients satisfy limn!±1 jcn(f)j= 0. � Parseval's identity states that f 2 L 2 (T) if and only if fcn(f)g 2 ` 2 (Z). � For 1 2 and f 2 L p (T), then f 2 L 2 (T) and fcn(f)g 2 ` 2 (Z).