Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums

TL;DR

This paper proposes new direct $L^2$ bounds for maximal partial sums of Walsh-Fourier series, focusing on dyadic partial sums, and introduces methods to improve understanding of these bounds without interpolation.

Contribution

It introduces a novel approach to obtain direct $L^2$ estimates for linearized partial sums in Walsh expansions, especially for dyadic partial sums, bypassing interpolation techniques.

Findings

Developed a method for direct $L^2$ bounds for dyadic partial sums

Outlined an approach for bounds on general linearized partial sums

Enhanced understanding of Walsh-Fourier series partial sums

Abstract

We outline an approach to obtain direct $L^2$ estimates not requiring interpolation for so-called linearized partial sums operators associated with expansions in Walsh functions. We focus specifically on a simpler case of dyadic partial sums but also outline a second approach to proving bounds on general linearized partial sums.