Smoothing the Nonsmoothness

TL;DR

This paper introduces a method to approximate nonsmooth functions with smooth ones, facilitating theoretical analysis, and demonstrates its effectiveness through approximation rates and practical applications.

Contribution

It proposes a sequence of smooth functions to approximate nonsmooth functions, providing a rate of approximation and illustrating its application.

Findings

Established approximation rates for smooth approximations

Demonstrated practical application of the approximation method

Facilitated theoretical analysis of nonsmooth functions

Abstract

To tackle difficulties for theoretical studies in situations involving nonsmooth functions, we propose a sequence of infinitely differentiable functions to approximate the nonsmooth function under consideration. A rate of approximation is established and an illustration of its application is then provided.