Norm Inequalities for the Fourier Coefficients of Some Almost Periodic Functions

TL;DR

This paper establishes norm inequalities for Fourier coefficients of Besicovitch almost periodic functions by leveraging Fefferman's embedding and interpolation theorems, extending classical inequalities to a broader class of functions.

Contribution

It introduces a novel approach to derive Paley-type inequalities for almost periodic functions using measure space embeddings and interpolation techniques.

Findings

Proves norm inequalities for Fourier coefficients of Besicovitch almost periodic functions.

Extends Paley's inequality to almost periodic functions.

Demonstrates the applicability of measure space embeddings in harmonic analysis.

Abstract

Using C. Fefferman's embedding of a charge space in a measure space allows us to apply standard interpolation theorems to prove norm inequalities for Besicovitch almost periodic functions. This yields an analogue of Paley's Inequality for the Fourier coefficients of periodic functions.