On the Convergence of a Weak Greedy Algorithm for the Multivariate Haar Basis

TL;DR

This paper introduces a family of weak greedy algorithms for the multivariate Haar basis in L1 spaces, proving their convergence and boundedness for all functions in these spaces.

Contribution

It defines and analyzes a new class of weak greedy algorithms for multivariate Haar basis approximation in L1 spaces, establishing their convergence and boundedness.

Findings

Proves convergence of the weak greedy algorithms.

Establishes uniform boundedness of the approximants.

Applicable to all functions in L1[0,1]^d.

Abstract

We define a family of weak thresholding greedy algorithms for the multivariate Haar basis for $L_1[0,1]^d$ ($d \ge 1$). We prove convergence and uniform boundedness of the weak greedy approximants for all $f \in L_1[0,1]^d$.