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- Title
A New Look at Old Theorems of Fejér and Hardy.
- Authors
Mikhailets, Vladimir; Murach, Aleksandr; Tsyhanok, Oksana
- Abstract
The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro 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 Fejér theorem on the uniform Cesàro 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.
- Subjects
TAUBERIAN theorems; INTERPOLATION spaces; BANACH spaces; NORMED rings; HOMOGENEOUS spaces; FOURIER series; SUMMABILITY theory
- Publication
Results in Mathematics / Resultate der Mathematik, 2024, Vol 79, Issue 2, p1
- ISSN
1422-6383
- Publication type
Article
- DOI
10.1007/s00025-023-02114-y