An inequality for the entropy numbers and its application

TL;DR

This paper establishes a new inequality relating entropy numbers and nonlinear Kolmogorov widths, providing bounds for m-term approximation algorithms, with implications for approximation theory.

Contribution

It introduces a novel inequality connecting entropy numbers to nonlinear Kolmogorov widths and applies it to m-term approximation with non-dictionary systems.

Findings

Derived an inequality for entropy numbers using nonlinear Kolmogorov widths.

Provided upper bounds for m-term approximation via the Weak Relaxed Greedy Algorithm.

Extended approximation bounds to systems that are not dictionaries.

Abstract

We prove an inequality for the entropy numbers in terms of nonlinear Kolmogorov's widths. This inequality is in a spirit of known inequalities of this type and it is adjusted to the form convenient in applications for $m$-term approximations with respect to a given system. Also, we obtain upper bounds for the $m$-term approximation by the Weak Relaxed Greedy Algorithm with respect to a system which is not a dictionary.