CBMS Regional Conference Series in Mathematics 1991; 132 pp; softcover Number: 79 Reprint/Revision History: reprinted 1997 ISBN10: 0821807315 ISBN13: 9780821807316 List Price: US$50 Member Price: US$40 All Individuals: US$40 Order Code: CBMS/79
 LittlewoodPaley theory was developed to study function spaces in harmonic analysis and partial differential equations. Recently, it has contributed to the development of the \(\varphi\)transform and wavelet decompositions. Based on lectures presented at the NSFCBMS Regional Research Conference on Harmonic Analysis and Function Spaces, held at Auburn University in July 1989, this book is aimed at mathematicians, as well as mathematically literate scientists and engineers interested in harmonic analysis or wavelets. The authors provide not only a general understanding of the area of harmonic analysis relating to LittlewoodPaley theory and atomic and wavelet decompositions, but also some motivation and background helpful in understanding the recent theory of wavelets. The book begins with some simple examples which provide an overview of the classical LittlewoodPaley theory. The \(\varphi\)transform, wavelet, and smooth atomic expansions are presented as natural extensions of the classical theory. Finally, applications to harmonic analysis (CalderónZygmund operators), signal processing (compression), and mathematical physics (potential theory) are discussed. Reviews "This monograph is an important and welcome addition to the growing literature in this area."  Mathematical Reviews "Useful for graduate students and researchers with interest in function spaces, approximation theory or wavelet theory."  Zentralblatt MATH Table of Contents  Calderón's formula and a decomposition of \(L^2(\mathbb R^n)\) ;
 Decomposition of Lipschitz spaces;
 Minimality of \(\dot B^0,1_1\) ;
 LittlewoodPaley theory;
 The Besov and TriebelLizorkin spaces;
 The \(\varphi\) transform;
 Wavelets;
 CalderónZygmund operators;
 Potential theory and a result of MuckenhouptWheeden;
 Further applications.
