Bernuau spline wavelets and Sturmian sequences

TL;DR

This paper introduces a novel spline wavelet construction supported by aperiodic, self-similar tilings of the real line, utilizing Sturmian sequences and analyzing specific cases like the Fibonacci chain.

Contribution

It presents a new method for constructing C^n spline wavelets based on self-similar tilings and Sturmian sequences, including explicit examples and potential multidimensional extensions.

Findings

Explicit construction of wavelets for Fibonacci and quadratic Pisot-Vijayaraghavan units

Analytic forms of scaling functions and wavelets as second-order splines

Insights into multidimensional spline wavelet construction using tilings

Abstract

A spline wavelets construction of class C^n(R) supported by sequences of aperiodic discretizations of R is presented. The construction is based on multiresolution analysis recently elaborated by G. Bernuau. At a given scale, we consider discretizations that are sets of left-hand ends of tiles in a self-similar tiling of the real line with finite local complexity. Corresponding tilings are determined by two-letter Sturmian substitution sequences. We illustrate the construction with examples having quadratic Pisot-Vijayaraghavan units (like tau = (1 + sqrt{5})/2 or tau^2 = (3 + sqrt{5})/2) as scaling factor. In particular, we present a comprehensive analysis of the Fibonacci chain and give the analytic form of related scaling functions and wavelets as splines of second order. We also give some hints for the construction of multidimensional spline wavelets based on stone-inflation tilings in arbitrary dimension.