Besov spaces of self-affine lattice tilings and pointwise regularity

Abstract

We investigate Besov spaces of self-affine tilings of ${\mathbb{R}}^n$ and discuss various characterizations of those Besov spaces. We see what is a finite set of functions which generates the Besov spaces from a view of multiresolution approximation on self-affine lattice tilings of ${\mathbb{R}}^n$. Using this result we give a generalization of already known characterizations of Besov spaces given by wavelet expansion and we apply to study the pointwise H$\ddot{\mathrm{o}}$lder space. Furthermore we give descriptions of scaling exponents measured by Besov spaces, and estimations of a pointwise H$\ddot{\mathrm{o}}$lder exponent to compute the pointwise scaling exponent of several oscillatory functions.