Besov spaces of self-affine lattice tilings and pointwise regularity

TL;DR

This paper explores Besov spaces on self-affine lattice tilings of Euclidean space, providing new characterizations, connections to wavelet expansions, and applications to pointwise regularity and oscillatory functions.

Contribution

It introduces a novel multiresolution framework for Besov spaces on self-affine tilings and extends existing characterizations via wavelet expansions to this setting.

Findings

Characterization of Besov spaces using a finite generating set of functions.

Generalization of wavelet-based characterizations of Besov spaces.

Estimation of pointwise Hölder exponents for oscillatory functions.

Abstract

We investigate Besov spaces of self-affine tilings of ${\Bbb R}^{n}$ 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 ${\Bbb R}^{n}$. 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${\ddot {\rm o}}$lder space. Furthermore we give descriptions of scaling exponents measured by Besov spaces, and estimations of a pointwise H${\ddot {\rm o}}$lder exponent to compute the pointwise scaling exponent of several oscillatory functions.