On Fourier re-expansions

TL;DR

This paper investigates the conditions under which Fourier re-expansions of absolutely convergent sine and cosine series also converge absolutely, using relations between Fourier and Hilbert transforms.

Contribution

It extends classical results by establishing new links between Fourier transforms and Hilbert transforms for re-expansion problems.

Findings

Identifies conditions for absolute convergence of Fourier re-expansions.

Reveals relations between Fourier transforms and Hilbert transforms.

Provides theoretical insights into Fourier series re-expansion behavior.

Abstract

We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.