Optimal $C^\infty$-approximation of functions with exponentially or sub-exponentially integrable derivative

TL;DR

This paper extends Meyers-Serrin approximation results to smooth functions with derivatives that have exponential or sub-exponential integrability, focusing on energy convergence with weighted integrals.

Contribution

It introduces new approximation results for functions with derivatives exhibiting exponential or sub-exponential growth, including convergence of weighted energy integrals.

Findings

Established Meyers-Serrin type approximation results for such functions.

Proved convergence of weighted energy integrals involving exponential growth functions.

Extended classical approximation theory to functions with rapidly growing derivatives.

Abstract

We discuss Meyers-Serrin's type results for smooth approximations of functions $b=b(t,x):\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^m$, with convergence of an energy of the form \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n} w(t,x) \varphi\left(|Db(t,x)|\right)\mathrm{d} x \mathrm{d} t\,, \] where $w>0$ is a suitable weight function, and $\varphi:[0,\infty)\to [0,\infty)$ is a convex function with $\varphi(0)=0$ having exponential or sub-exponential growth.