ADVAI\CES in communication, sensing, and computational power have Jed to an cxplosion of data. The size and varied formats for these datasets challenge existing techniqucs for transmission, storage, querying, display, and numerical manipula tion. This Ieads to the paradoxical situation where experiments or numerical com pulations produce rich, detailed inforrnation, for which, at this point, no adequate analysis tools exist. -Conference annow cement, Joint IDR-1/v A Workshop on Ideal Data Nepresentaticm, Minneapolis, R. De\'ore and A. Ron, cJ/gani ers Wavelct theory stands on the interface betwccn signal processing and harmonic analy sis, the rnathematical tools involved in digitizing continuous data with a vicw to storage, and thc synthesis proccss, recreating, for cxample, a picturc or time signal from stored data. The algorithms involved go under the name of tilter banks, and their spectacular efticiency derivcs in patt from the use of hidden self-similarity, relati\ c to somc scaling operation, in the daLJ. being analyzed. Observations or time signals are functions, and classes of functions make up linear spaces. Numcrical correlations add structure to thc spaccs at hand, Hilbcrt spaces. There are opcrators in the spaces deriving lrom the dis crcte data and others from the spaces of continuous signals. The first type arc good for computations, whilc the sccond retlect the real world. The operators between thc two are the focus of the prescnt monograph. Relations between operations in thc discrete xn Preface and continuous domains are studied as symbols."

Combines analysis and tools from probability, harmonic analysis, operator theory, and engineering (signal/image processing)

Interdisciplinary focus with hands-on approach, generous motivation and new pedagogical techniques

Numerous exercises reinforce fundamental concepts and hone computational skills

Separate sections explain engineering terms to mathematicians and operator theory to engineers

Fills a gap in the literature

This text, combining analysis and tools from mathematical probability, focuses on a systematic and novel exposition of a recent trend in pure and applied mathematics. The emphasis is on the unity of basis constructions and their expansions (bases which are computationally efficient), and on their use in several areas: from wavelets to fractals. The aim of this book is to show how to use processes from probability, random walks on branches, and their path-space measures in the study of convergence questions from harmonic analysis, with particular emphasis on the infinite products that arise in the analysis of wavelets. The book brings together tools from engineering (especially signal/image processing) and mathematics (harmonic analysis and operator theory). audience of students and workers in a variety of fields, meeting at the crossroads where they merge; hands-on approach with generous motivation; new pedagogical features to enhance teaching techniques and experience; includes more than 34 figures with detailed captions, illustrating the main ideas and visualizing the deeper connections in the subject; separate sections explain engineering terms to mathematicians and operator theory to engineers; and, interdisciplinary presentation and approach, combining central ideas from mathematical analysis (with a twist in the direction of operator theory and harmonic analysis), probability, computation, physics, and engineering. The presentation includes numerous exercises that are essential to reinforce fundamental concepts by helping both students and applied users practice sketching functions or iterative schemes, as well as to hone computational skills. Graduate students, researchers, applied mathematicians, engineers and physicists alike will benefit from this unique work in book form that fills a gap in the literature.

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.