On the divergence of greedy algorithms with respect to Walsh subsystems in $L$

TL;DR

This paper demonstrates that there exists an $L^1$ function for which a greedy algorithm based on Walsh subsystems fails to converge in $L^1$ norm, showing limitations of Walsh bases in greedy approximation.

Contribution

It proves the existence of an $L^1$ function where Walsh-based greedy algorithms do not converge, highlighting a divergence in greedy approximation methods.

Findings

Existence of a non-converging $L^1$ function for Walsh greedy algorithms

Walsh subsystem is not a quasi-greedy basis in $L^1$

Limitations of Walsh bases in greedy approximation methods

Abstract

In this paper we prove that there exists a function which $f(x)$ belongs to $L^1[0,1]$ such that a greedy algorithm with regard to the Walsh subsystem does not converge to $f(x)$ in $L^1[0,1]$ norm, i.e. the Walsh subsystem $\{W_{n_k}\}$ is not a quasi-greedy basis in its linear span in $L^1$.