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The subject of this book stands at the crossroads of ergodic theory and measurable dynamics. With an emphasis on irreversible systems, the text presents a framework of multi-resolutions tailored for the study of endomorphisms, beginning with a systematic look at the latter. This entails a whole new set of tools, often quite different from those used for the “easier” and well-documented case of automorphisms. Among them is the construction of a family of positive operators (transfer operators), arising naturally as a dual picture to that of endomorphisms. The setting (close to one initiated by S. Karlin in the context of stochastic processes) is motivated by a number of recent applications, including wavelets, multi-resolution analyses, dissipative dynamical systems, and quantum theory. The automorphism-endomorphism relationship has parallels in operator theory, where the distinction is between unitary operators in Hilbert space and more general classes of operators such as contractions. There is also a non-commutative version: While the study of automorphisms of von Neumann algebras dates back to von Neumann, the systematic study of their endomorphisms is more recent; together with the results in the main text, the book includes a review of recent related research papers, some by the co-authors and their collaborators.

Mathematics. --- Mathematical statistics. --- Functional analysis. --- Measure theory. --- Operator theory. --- Probabilities. --- Thermodynamics. --- Measure and Integration. --- Functional Analysis. --- Probability and Statistics in Computer Science. --- Probability Theory and Stochastic Processes. --- Operator Theory. --- Chemistry, Physical and theoretical --- Dynamics --- Mechanics --- Physics --- Heat --- Heat-engines --- Quantum theory --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Functional analysis --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Math --- Science --- Statistical methods --- Computer science. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Informatics --- Endomorphisms (Group theory) --- Group theory

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Mathematical analysis --- Numerical solutions of algebraic equations --- Infinite matrices. --- Transformations (Mathematics) --- Iterative methods (Mathematics) --- Matrices infinies --- Transformations (Mathématiques) --- Itération (Mathématiques) --- 51 <082.1> --- Mathematics--Series --- Transformations (Mathématiques) --- Itération (Mathématiques)

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Motivated by applications, an underlying theme in analysis is that of finding bases and understanding the transforms that implement them. These may be based on Fourier techniques or involve wavelet tools; they may be orthogonal or have redundancies (e.g., frames from signal analysis). Representations, Wavelets, and Frames contains chapters pertaining to this theme from experts and expositors of renown in mathematical analysis and representation theory. Topics are selected with an emphasis on fundamental and timeless techniques with a geometric and spectral-theoretic flavor. The material is self-contained and presented in a pedagogical style that is accessible to students from both pure and applied mathematics while also of interest to engineers. The book is organized into five sections that move from the theoretical underpinnings of the subject, through geometric connections to tilings, lattices and fractals, and concludes with analyses of computational schemes used in communications engineering. Within each section, individual chapters present new research, provide relevant background material, and point to new trends and open questions. Contributors: C. Benson, M. Bownik, V. Furst, V.W. Guillemin, B. Han, C. Heil, J.A. Hogan, P.E.T. Jorgensen, K. Kornelson, J.D. Lakey, D.R. Larson, K.D. Merrill, J.A. Packer, G. Ratcliff, K. Shuman, M.-S. Song, D.W. Stroock, K.F. Taylor, E. Weber, X. Zhang.

Wavelets (Mathematics) --- Frames (Combinatorial analysis) --- Representations of Banach modules. --- Banach modules (Algebra) --- Combinatorial designs and configurations --- Wavelet analysis --- Harmonic analysis --- Fourier analysis. --- Harmonic analysis. --- Mathematics. --- Functional analysis. --- Mathematical Modeling and Industrial Mathematics. --- Fourier Analysis. --- Signal, Image and Speech Processing. --- Abstract Harmonic Analysis. --- Applications of Mathematics. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Analysis, Fourier --- Mathematical models. --- Signal processing. --- Image processing. --- Speech processing systems. --- Applied mathematics. --- Engineering mathematics. --- 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 --- Engineering --- Engineering analysis --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Models, Mathematical --- Simulation methods