Derivation of an upper bound of the constant in the error bound for a near best m-term approximation

TL;DR

This paper derives an explicit upper bound for the constant in the error estimate of near best m-term approximations using Haar-like bases in L^p spaces, improving understanding of approximation quality.

Contribution

It provides the first explicit upper bound for the constant in the error bound for near best m-term approximation with Haar systems in L^p spaces.

Findings

Established an explicit upper bound for the constant C(p)

Enhanced the theoretical understanding of approximation errors in L^p spaces

Improved bounds can inform practical approximation algorithms

Abstract

In the paper "The best m-term approximation and greedy algorithms" (V. N. Temlyakov), an error bound for a near best m-term approximation of a function g in L^p([0,1]^d) is provided, using a basis L^p-equivalent to the Haar system, where p is greater than one and less than infinity and d is a natural number. The bound includes a constant C(p) that is not given explicitly. The goal of this paper is to find an upper bound of the constant for the Haar system.