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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.
Content Level »Research
Keywords »Approximation - Homotopy - Signal - Winding number - algorithms - approximation theory - calculus - chaos - chaos theory - communication - computer - computer graphics - filters - homotopy theory - signal processing
Preface * List of Figures * List of Tables * Introduction * Homotopy Theory and Cascades * Can You Hear the Shape of a Wavelet? * The Transfer Operator and Perron-Frobenius Theory * The Fixed Points of the Transfer Operator * Orthogonalization and Isospectral Approximation * References * Index