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This book, combining analysis and tools from mathematical probability, focuses on a systematic and novel presentation of recent trends in pure and applied mathematics: the emergence of three fields, wavelets, signals and fractals. The unity of basis constructions and their expansions is emphasized as the starting point for the development of bases that are computationally efficient for use in several areas from wavelets to fractals. The book brings together tools from engineering and math, especially from signal- and image processing, and from harmonic analysis and operator theory. The presentation is aimed at graduate students, as well as users from a diverse spectrum of applications. Key features: • A hands-on approach for students, including tutorials and numerous exercises; • Excellent motivation throughout; • New pedagogical features: glossary of terms, their use in mathematics and in engineering, help for cross-audiences, image processing, visual presentation of key algorithms, structure and geometry of big matrix computations, explanation of uses of the theory in applications outside of mathematics; • Includes more than 50 figures with captions, illustrating the main ideas, plus engineering diagrams, graphic renditions of algorithms, and separate illustrations; • Separate sections in the book explain engineering terms to mathematicians, and operator theory to engineers; • Each chapter concludes with a helpful guide to the literature allowing students to follow up on the topics in the book. Palle E.T. Jorgensen is a Professor of Mathematics at the University of Iowa. This book is based in part on interdisciplinary courses that he has taught over the last several years, and on his work with his current and former students. His most recent book was written jointly with Ola Bratteli and is entitled Wavelets through a Looking Glass, ©2002 Birkhäuser Boston.

Fractals. --- Mathematical analysis. --- Probabilities. --- Wavelets (Mathematics) --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- 517.1 Mathematical analysis --- Mathematical analysis --- Wavelet analysis --- Harmonic analysis --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Probabilities --- Fractals --- Distribution (Probability theory. --- Global analysis (Mathematics). --- Mathematics. --- Engineering mathematics. --- Probability Theory and Stochastic Processes. --- Analysis. --- Applications of Mathematics. --- Mathematical and Computational Engineering. --- Signal, Image and Speech Processing. --- Information and Communication, Circuits. --- Engineering --- Engineering analysis --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Distribution functions --- Frequency distribution --- Characteristic functions --- Harmonic analysis. --- Analysis (Mathematics). --- Applied mathematics. --- Signal processing. --- Image processing. --- Speech processing systems. --- Information theory. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Communication theory --- Communication --- Cybernetics --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Computer science --- Probability Theory. --- Mathematical and Computational Engineering Applications. --- Signal, Speech and Image Processing . --- Mathematical Applications in Computer Science. --- Data processing. --- Computer mathematics --- Electronic data processing