The Haar System in Besov-type Spaces

TL;DR

This paper investigates the characterization of Besov-type spaces using Fourier--Haar coefficients, examining parameter restrictions and the regularity of characteristic functions within these spaces.

Contribution

It provides necessary parameter restrictions for Fourier--Haar coefficient characterizations and analyzes the regularity of characteristic functions in Besov-type spaces.

Findings

Identifies parameter conditions for Fourier--Haar characterization

Measures regularity of characteristic functions in Besov spaces

Establishes conditions for linear functional continuity

Abstract

Some Besov-type spaces $B^{s,\tau}_{p,q}(\mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $\tau$, $p$, $q$ and $n$ of this characterization. Therefore, the authors measure the regularity of the characteristic function $\mathcal X$ of the unit cube in $\mathbb{R}^n$ via the Besov-type spaces $B^{s,\tau}_{p,q}(\mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $\langle f, \mathcal{X} \rangle$ generates a continuous linear functional on $B^{s,\tau}_{p,q}(\mathbb{R}^n)$.