Wavelets, Functions, and Operators
Philippe Tchamitchian · 1996
Abstract The development of wavelets has many sources which certainly ex plains in part the rapid growth of the field, and its success in various scientific disciplines. These notes aim to provide the reader with an introduction to the basic mathematical concepts of wavelets which, until now, govern almost all the applications of wavelets to the study of operators. Before delving into the study of operators, we first concentrate on functions, or, from a more practical point of view, on signals. The first part of the notes is mainly concerned with the interpretation of wavelet coefficients. Our tutorial is far from complete, and in particular we avoid issues of practical implementation. More generally, we do not consider the important subject of filtering which has recently lead to the remarkable concept of wavelets packets (Wickerhauser 1993) which essentially allows the wavelet transform to be embedded in a much larger family of transformations. Regarding this recent development, “standard” wavelet theory may appear already out dated or too restrictive. We think however that a solid grounding of the basic wavelet transform will provide a strong foundation for all related disciplines for a long time.