Periodic Splines, Harmonic Analysis and Wavelets
نویسنده
چکیده
We discuss here wavelets constructed from periodic spline functions. Our approach is based on a new computational technique named Spline Harmonic Analysis (SHA). SHA to be presented is a version of harmonic analysis operating in the spaces of periodic splines of defect 1 with equidistant nodes. Discrete Fourier Transform is a special case of SHA. The continuous Fourier Analysis is the limit case of SHA as the degree of splines involved tends to infinity. Thus SHA bridges the gap between the discrete and the continuous versions of the Fourier Analysis. SHA can be regarded as a computational version of the harmonic analysis of continuous periodic functions from discrete noised data. SHA approach to wavelets yields a tool just as for constructing a diversity of spline wavelet bases, so for a fast implementation of the decomposition of a function into a fitting wavelet representation and its reconstruction. Via this approach we are able to construct wavelet packet bases for refined frequency resolution of signals. In the paper we present also algorithms for digital signal processing by means of spline wavelets and wavelet packets. The algorithms established are embodied in a flexible multitasking software for digital signal processing. §
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