The Critical Smoothness of Generalized Functions

TL;DR

This paper characterizes how the critical smoothness of periodic generalized functions varies with integrability parameters and links this to their compressibility in wavelet bases, advancing understanding of function space properties.

Contribution

It provides a complete characterization of the possible critical smoothness functions for generalized periodic functions and relates this to their wavelet compressibility.

Findings

Explicit description of all possible critical smoothness functions

Characterization of function compressibility in wavelet bases

Insight into the relationship between smoothness and integrability

Abstract

For each integrability parameter $p \in (0,\infty]$, the critical smoothness of a periodic generalized function $f$, denoted by $s_f(p)$ is the supremum over the smoothness parameters $s$ for which $f$ belongs to the Besov space $B_{p,p}^s$ (or other similar function spaces). This paper investigates the evolution of the critical smoothness with respect to the integrability parameter $p$. Our main result is a simple characterization of all the possible critical smoothness functions $p\mapsto s_f(p)$ when $f$ describes the space of generalized periodic functions. We moreover characterize the compressibility of generalized periodic functions in wavelet bases from the knowledge of their critical smoothness function.