REGULAR ALMOST-PERIODIC FUNCTIONS OF LIMITED VARIATION

Abstract

This article investigates the main properties of regular almost-periodic functions of bounded variation and the convergence problems of their Fourier series. The primary objective of the study is to determine sufficient conditions for the convergence of Fourier series associated with regular almost-periodic functions whose Fourier coefficients diverge to positive infinity. The authors begin by presenting rigorous mathematical definitions of almost-periodic functions, Fourier series, and bounded variation. Subsequently, appropriate sequences are constructed for these functions, and their convergence properties are analyzed.

The principal result of the paper is formulated in Theorem 1. It is proven that if a periodic-type function has bounded variation and a given sequence converges, then the sequence derived from it also converges. The proof is based on classical mathematical tools, including the mean value theorem, Hölder’s inequality, Cauchy’s inequality, and Abel’s transformation. The results obtained are of significant theoretical importance in the field of Fourier analysis and contribute to a deeper understanding of almost-periodic functions within functional analysis. Moreover, the findings provide a solid theoretical foundation for further research in this area.