On the almost everywhere and norm convergences of N\"orlund means with respect to Vilenkin systems

TL;DR

This paper investigates the convergence properties of Nörlund means with monotone coefficients in Vilenkin-Fourier series, establishing almost everywhere and norm convergence results within modern harmonic analysis.

Contribution

It extends the theory of Nörlund means by proving convergence in Lebesgue and Vilenkin-Lebesgue points for Vilenkin-Fourier series with monotone coefficients.

Findings

Proved almost everywhere convergence of Nörlund means.

Established norm convergence in Lebesgue spaces.

Connected convergence results to modern harmonic analysis

Abstract

Unlike the classical theory of Fourier series which deals with decomposition of a function into sinusoidal waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series but there are a lot of differences also. The aim of my master thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate N\"orlund means but only in the case when their coefficients are monotone and prove convergence in Lebesgue and Vilenkin-Lebesgue points. Since almost everywhere points are Lebesgue and Vilenkin-Lebesgue points for any integrable functions we obtain almost everywhere convergence of such summability methods.