Function Spaces and Wavelets on Domains
This book develops a theory of wavelet bases and frames for function spaces on various types of domains (such as Euclidean n-spaces) and on related n-manifolds. Basic notation and classical results are repeated in order to make the text self-contained. The book is organised as follows. Chapter 1 deals with the usual spaces on Rn, periodic spaces on Rn and on the n-torus Tn, and their wavelet expansions under natural restrictions for the parameters involved. Spaces on arbitrary domains are discussed in chapter 2. The heart of the exposition is found in chapters 3 and 4, where the author develops the theory of function spaces on so-called thick domains (including wavelet expansions and extensions to corresponding spaces on Rn). In chapter 5 this is completed with spaces on smooth manifolds and smooth domains. In the final chapter, the author discusses desirable properties of wavelet expansions in function spaces (introducing the notation of Riesz wavelet bases and frames). This chapter also deals with some related topics, in particular with spaces on cellular domains. The book is addressed to two types of reader: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions, and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of reader, the preface contains a guide to where one will find basic definitions and key assertions.