The Haar system in Besov-type spaces
Volume 253 / 2020
Studia Mathematica 253 (2020), 129-162
MSC: Primary 42C15; Secondary 46E35.
DOI: 10.4064/sm180828-9-7
Published online: 27 January 2020
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 on the parameters $s$, $\tau $, $p$, $q$ and $n$ in order to have such a characterization. To do so, the authors measure the regularity of the characteristic function $\mathcal X$ of the unit cube in $\mathbb {R}^n$ via Besov-type spaces $B^{s,\tau }_{p,q}(\mathbb {R}^n)$. Furthermore, the authors study necessary and sufficient conditions for the operation $\langle f, \mathcal {X} \rangle $ to generate a continuous linear functional on $B^{s,\tau }_{p,q}(\mathbb {R}^n)$.
