Absolute convergence factors of Lipshitz class functions for general Fourier series

TL;DR

This paper investigates conditions on sequences that ensure absolute convergence of Fourier series for Lip1 class functions, identifying optimal criteria for orthonormal systems.

Contribution

It introduces new conditions on sequences that serve as absolute convergence factors for Fourier series of Lip1 functions, proving their optimality.

Findings

Identified sequences that guarantee absolute convergence of Fourier series.

Established necessary and sufficient conditions for orthonormal systems.

Proved the optimality of the derived convergence conditions.

Abstract

The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions $f\in$ Lip1 class provides absolute convergence of Fourier series. In particular we found special conditions for the functions of orthonormal system, for which the above sequences are absolute convergence factors of Fourier series of functions of Lip1 class. It is established that the resulting conditions are best possible in certain sense.