Approximation of Lipschitz functions by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

TL;DR

This paper demonstrates that on certain Banach spaces, uniformly continuous functions can be approximated uniformly by Lipschitz, C^{p} smooth functions, extending classical approximation results to a Lipschitz context.

Contribution

It establishes a Lipschitz approximation theorem for uniformly continuous functions on weakly compactly generated Banach spaces with smooth norms, correcting and improving previous results.

Findings

Uniform approximation of continuous functions by Lipschitz, C^{p} smooth functions.

Approximation of Lipschitz functions with controlled Lipschitz constants.

Extension of classical approximation theorems to a Lipschitz framework.

Abstract

This note corrects a gap and improves results in an earlier paper by the first named author. More precisely, it is shown that on weakly compactly generated Banach spaces X which admit a C^{p} smooth norm, one can uniformly approximate uniformly continuous functions f:X->R by Lipschitz, C^{p} smooth functions. Moreover, there is a constant C>1 so that any L-Lipschitz function f:X->R can be uniformly approximated by CL-Lipschitz, C^{p} smooth functions. This provides a `Lipschitz version' of the classical approximation results of Godefroy, Troyanski, Whitfield and Zizler.