Analysis and Probability: Wavelets, Signals, Fractals
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. —John von Neumann While this is a course in analysis, our approach departs from the beaten path in some ways. Firstly, we emphasize a variety of connections to themes from neighboring fields, such as wavelets, fractals and signals; topics typically not included in a gradu ate analysis course. This in turn entails excursions into domains with a probabilistic flavor. Yet the diverse parts of the book follow a common underlying thread, and to gether they constitute a good blend; each part in the mix naturally complements the other. In fact, there are now good reasons for taking a wider view of analysis, for ex ample the fact that several applied trends have come to interact in new and exciting ways with traditional mathematical analysis—as it was taught in graduate classes for generations. One consequence of these impulses from "outside" is that conventional boundaries between core disciplines in mathematics have become more blurred. Fortunately this branching out does not mean that students will need to start out with any different or additional prerequisites. In fact, the ideas involved in this book are intuitive, natural, many of them visual, and geometric. The required background is quite minimal and it does not go beyond what is typically required in most graduate programs.
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adjoint applications BrJo02b C*-algebra Cantor set Chapter cocycle coefficients computation convergence corresponding Cuntz algebra Cuntz relations Daubechies defined denote DuJo06b dyadic endomorphism engineering equivalent example Exercise Figure finite formula Fourier fractals frame function h Geometry Haar measure Haar wavelet Haar's harmonic analysis harmonic function Hausdorff Hilbert space idea image processing infinite products integers isometries iterated function systems Kolmogorov lattice Lebesgue measure Lemma linear low-pass mapping martingales Math mathematics matrix measures Px multiresolution operator theory orthogonal orthonormal basis P.E.T. Jorgensen Parseval path-space measures probability space Proof random walk reader recursive representation result Ruelle operator satisfies scale-similarity scaling function scaling identity Section sequence signal processing space H subband filters subset subspaces tensor products Theorem transfer operator transform transition probabilities unitary operator vector wavelet algorithms wavelet bases wavelet constructions wavelet packets
Page vii - If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
Page 246 - Mallat, A wavelet tour of signal processing, Academic Press: San Diego, 1998. 19. M. Popescu, B. Nicula, P. Cristea, and A. Bezerianos, "High resolution ECG filtering using wavelet shrinkage technique,
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