Bernuau Spline Wavelets andSturmian Sequences

We present spline wavelets of class Cⁿ(R) supported by sequences of aperiodicdiscretizations of R. The construction is based on multiresolution analysis recently elaboratedby G. Bernuau. At a given scale, we consider discretizations that are sets of left-hand ends oftiles in a self-similar tiling of the real line with finite local complexity. Corresponding tilingsare determined by two-letter Sturmian substitution sequences. We illustrate the construction withexamples 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 andgive the analytic form of related scaling functions and wavelets. We also give some hints forthe construction of multidimensional spline wavelets based on stone-inflation tilings in arbitrarydimension.