On the generalized Approximate Weak Chebyshev Greedy Algorithm

TL;DR

This paper analyzes a flexible version of the Weak Chebyshev Greedy Algorithm in Banach spaces, establishing convergence conditions with error tolerances and demonstrating divergence in certain smooth spaces.

Contribution

It introduces a generalized greedy algorithm allowing imprecise steps and provides sharp convergence criteria in uniformly smooth Banach spaces.

Findings

Convergence guaranteed under specific error conditions

Conditions are proven to be sharp

Constructed example of divergence in a smooth Banach space

Abstract

In this paper we study greedy approximation in Banach spaces. We discuss a modification of the Weak Chebyshev Greedy Algorithm, in which steps of the algorithm can be executed imprecisely. Such inaccuracies are represented by both absolute and relative errors, which provides more flexibility for numerical applications. We obtain the conditions on the error parameters that guarantee convergence of the algorithm in all uniformly smooth Banach spaces and prove that these conditions are sharp. Additionally, we discuss the conditions that need to be imposed on a Banach space to guarantee convergence of the Chebyshev Greedy Algorithm, and construct a smooth Banach space in which the algorithm diverges.