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Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form. Key features of this self-contained monograph include: • fine exposition covering the history of the subject • up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis • presentation of DRE techniques using a broad array of examples • good balance between theory and application • coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals • excellent and comprehensive bibliography and index This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

This volume contains papers selected from the Wavelet Analysis and Multiresolution Methods Session of the AMS meeting held at the University of Illinois at Urbana-Champaign. The contributions cover: construction, analysis, computation and application of multiwavelets; scaling vectors; nonhomogenous refinement; mulivariate orthogonal and biorthogona