Computing derivatives of scaling functions and wavelets

Abstract

This paper provides a general approach to the computation, for sufficiently regular multiresolution analyses, of scaling functions and wavelets and their derivatives. Two distinct iterative schemes are used to determine the multiresolution functions, the so-called 'cascade' algorithm and an eigenvector-based method. We present a novel development of these procedures which not only encompasses both algorithms simultaneously but also applies to the computation of derivatives of the functions. With this we demonstrate that the differences between the two algorithms are due solely to their respective initializations. We prove that the cascade initialization can be used only to compute the functions themselves, while the eigenvector one works for their derivatives as well. Finally, as an alternative to the results of Daubechies and Lagarias (1991, 1992), we derive a new, simpler normalization formula for the eigenvector method.