Multiresolution Analyses on Quasilattices

TL;DR

This paper explores the construction of multiresolution analyses on quasilattices using Pisot-Vijayaraghavan numbers, introducing new refinable distributions and functions with potential multidimensional applications.

Contribution

It introduces a novel framework for multiresolution analysis on quasilattices using PV numbers, including new refinable distributions and functions, and discusses multidimensional extensions.

Findings

Relations between Fourier moduli and polynomials established

Constructed families of quasilattices and refinable distributions

Conjecture that scalar refinable distributions are never integrable

Abstract

We derive relations between geometric means of the Fourier moduli of a refinable distribution and of a related polynomial. We use Pisot-Vijayaraghavan numbers to construct families of one dimension quasilattices and multiresolution analyses spanned by distributions that are refinable with respect to dilation by the PV numbers and translation by quasilattice points. We conjecture that scalar valued refinable distributions are never integrable, construct piecewise constant vector valued refinable functions, and discuss multidimensional extensions.