Optimal convergence factors for general Fourier coefficients

Larry Gogoladze; Giorgi Cagareishvili

doi:10.1515/gmj-2023-2017

Article

Optimal convergence factors for general Fourier coefficients

Larry Gogoladze and Giorgi Cagareishvili

Published/Copyright: March 29, 2023

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From the journal Georgian Mathematical Journal Volume 30 Issue 4

Abstract

Stefan Banach proved that even the Fourier series of the function f ⁢ ( x ) = 1 ( x ∈ [ 0 , 1 ] ) might not be convergent for some orthonormal systems (ONS). Thus we can conclude that the Fourier series of functions belonging to a certain differential class cannot be convergent in general. On the other hand, for classical ONS (trigonometric, Haar, Walsh systems, …), the convergence of Fourier series of differentiable class functions is a simple subject. In this paper, we investigate the sequence of positive numbers such that, multiplying the Fourier coefficients of Lipschitz class functions by these numbers, one obtains a convergent series of the special form. From the convergence of these special form series, we derive the convergence of the special Fourier series of Lipschitz class functions with respect to general ONS. The obtained results are best possible.

Keywords: General Fourier coefficients; convergence factors; Lipschitz class functions; Banach space

MSC 2010: 42C10

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Received: 2022-09-01

Revised: 2022-10-24

Accepted: 2022-10-28

Published Online: 2023-03-29

Published in Print: 2023-08-01

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Articles in the same Issue

https://doi.org/10.1515/gmj-2023-2017

Keywords for this article

General Fourier coefficients; convergence factors; Lipschitz class functions; Banach space