2003/09/24 by Miroslav Andrle, Andrle, Miroslav, Cestmir Burdik +5
Biochemistry, Genetics and Molecular Biology · Materials Science · Mathematics · Physics and Astronomy · #42C40 #52C23 #65D07 #65T60 #FOS: Physical sciences #Fractal and DNA sequence analysis #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties #math-ph #math.MP #msc:42C40 #msc:52C23 #msc:65D07 #msc:65T60
paper · pdf · doi:10.48550/arxiv.math-ph/0309054
41 pages,10 figures
arxiv created 2003/09/24 · openalex publication_date 2003/09/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A spline wavelets construction of class Cn(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 + sqrt5)/2 or tau2 = (3 + sqrt5)/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.