By Palle E. T. Jorgensen, B. Treadway
Combines research and instruments from probability, harmonic research, operator idea, and engineering (signal/image processing)
Interdisciplinary focus with hands-on method, beneficiant motivation and new pedagogical techniques
Numerous routines make stronger primary suggestions and hone computational skills
Separate sections clarify engineering phrases to mathematicians and operator thought to engineers
Fills a niche within the literature
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Extra info for Analysis and Probability Wavelets, Signals, Fractals
2 Path space A well-tested tool in analysis and mathematical physics centers around the application of path-space measures. This tool is used in attacking a variety of singular convergence, or approximation, problems. We will adopt this viewpoint in our study of wavelet approximations. Traditionally, the setting for wavelet questions has included assumptions concerning continuity, or some kind of differentiability. In contrast, we shall work almost entirely in the measurable category. One advantage of this approach is that we stay in the measurable category when addressing problems from multiresolution analysis (MRA).
Chapter 7 [Bri95, BrRo91, JaMROl] 2000 The International Standards Organization A wavelet-based picture compression standard, called JPEG 2000, for digital encoding of images. Chapters 3, 7, especially the figures [JaMROl] 1994 David Donoho statistics, mathematics Pioneered the use of wavelet bases and toolsfi*omstatistics to "denoise" images and signals. Chapter 7 [DoJo94, JaMRO 1 ] xxxiv Getting started General theory We consider a measurable space X with an endomorphism a, mapping onto X, such that the inverse image of every point is of the same finite cardinaUty.
Each measure Px depends on the initial point x and on a prescribed and fixed weight fimction W on X. We shall think of Px as a measure on (infinite) paths originating at x, but its construction rests on a fiindamental idea of Kolmogorov which states that Px is determined by its value on paths that are only fixed at a finite number n of places. , to Q, if and only if there is consistency from ^ to « + 1 for all n. 1 (Kolmogorov consistency, p. 46) for precise details. Our probability space Q will be the space of all infinite words co, Q will be given its standard Tychonofif infiniteproduct topology, and C (Q) will denote the space of all continuous complex-valued fijnctions on Q.
Analysis and Probability Wavelets, Signals, Fractals by Palle E. T. Jorgensen, B. Treadway