The Haar System in Besov-type Spaces

Some Besov-type spaces $B_{p,q}^{s,\tau }({\mathbb{R}}^n)$ 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 $B_{p,q}^{s,\tau }({\mathbb{R}}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $⟨f,\mathcal{X}⟩$ generates a continuous linear functional on $B_{p,q}^{s,\tau }({\mathbb{R}}^n)$.