Numerical integration without smoothness assumption

TL;DR

This paper explores numerical integration without assuming any smoothness of the functions, demonstrating how nonlinear approximation methods like greedy approximation can still ensure error decay rates.

Contribution

It introduces a framework for numerical integration in very general function classes without smoothness, leveraging nonlinear approximation techniques.

Findings

Error decay rates are achievable without smoothness assumptions.

Greedy approximation effectively guarantees integration accuracy.

Framework broadens applicability of numerical integration methods.

Abstract

We consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.