Bernuau Spline Wavelets and Sturmian Sequences

We present spline wavelets of class Cⁿ(R) supported by sequences of aperiodic discretizations of R. 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 = (1+\sqr{5})/2 or ² = (3+\sqr{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. We also give some hints for the construction of multidimensional spline wavelets based on stone-inflation tilings in arbitrary dimension.