Covering numbers for bounded variation functions

TL;DR

This paper investigates the minimal number of bounded variation functions needed to approximate any such function within a specified accuracy in the L^1 norm, providing bounds on this covering number.

Contribution

It introduces upper and lower bounds for covering numbers of bounded variation functions in the L^1 metric, advancing understanding of their approximation complexity.

Findings

Established bounds on covering numbers for bounded variation functions.

Quantified the approximation complexity in terms of epsilon.

Provided theoretical estimates for function representation.

Abstract

In this paper, we provide upper and lower estimates for the minimal number of functions needed to represent a bounded variation function with an accuracy of epsilon with respect to ${\bf L}^1$-distance.