On the divergence of subsequences of partial Walsh-Fourier sums

TL;DR

This paper identifies specific increasing sequences of natural numbers for which there exists a function in L[0,1) with Walsh-Fourier sums diverging everywhere along those sequences, and provides growth conditions for such functions.

Contribution

It introduces conditions on sequences and growth functions that guarantee the existence of functions with divergent Walsh-Fourier subsequences everywhere.

Findings

Existence of functions with divergent Walsh-Fourier subsequences for certain sequences.

Conditions on growth functions ensuring such divergence.

Characterization of sequences leading to divergence.

Abstract

A class of increasing sequences of natural numbers $(n_k)$ is found for which there exists a function $f\in L[0,1)$ such that the subsequence of partial Walsh-Fourier sums $(S_{n_k}(f))$ diverge everywhere. A condition for the growth order of a function $\varphi:[0,\infty)\rightarrow[0,\infty)$ is given fulfilment of which implies an existence of above type function $f$ in the class $\varphi(L)[0,1)$.