A new look at old theorems of Fej\'{e}r and Hardy

TL;DR

This paper introduces a new Tauberian theorem for Cesàro summable series in normed spaces, extending classical results of Fejér and Hardy to broader Banach space contexts and weaker conditions.

Contribution

It generalizes Hardy and Littlewood's results, providing weaker convergence conditions for Fourier series in homogeneous Banach spaces and exploring new examples and interpolation properties of these spaces.

Findings

Established a new Tauberian theorem for Cesàro summable series.

Generalized Fejér's theorem on Cesàro summability.

Extended Hardy's theorem on uniform convergence of Fourier series.

Abstract

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Ces\`{a}ro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy's condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fej\'{e}r theorem on the uniform Ces\`{a}ro summability of the Fourier series on sets by means of a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets.