Multiresolution Analysis and Supercompact Multiwavelets

Abstract

The Haar wavelets can represent exactly any piecewise constant function. The motivation for the present development is Alpert's family of compact orthogonal multiwavelets that can represent exactly any piecewise polynomial function. We choose to derive the algorithm in the style and notation of Harten's multiresolution analysis as extended to multiwavelets by the authors. We begin with a description of the nested grid hierarchy. Next comes the decomposition, which is the heart of the algorithm, and finally the reconstruction. The basis functions (which are nonfractal) retain the spatial compactness of the Haar basis functions, which enhances the algorithm application to nonperiodic and piecewise continuous data.