Strong boundedness, strong convergence and generalized variation

TL;DR

This paper investigates the conditions under which certain trigonometric series are Fourier series of functions in all L^p spaces, and presents new results on the strong convergence of Fourier series for functions with generalized bounded variation.

Contribution

It establishes that strongly bounded trigonometric series with specific coefficient conditions are Fourier series of functions in all L^p spaces and introduces new results on strong convergence for functions of generalized bounded variation.

Findings

Strong boundedness at two points implies the series is a Fourier series of an L^p function.

New convergence results for Fourier series of functions with generalized bounded variation.

Conditions on coefficients ensure the series' strong convergence.

Abstract

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $L^{p}$ spaces, $1\leq p < \infty$. We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation.