Wave Packets and Related Transforms
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Beschreibung
Time-frequency and time-scale analysis, two of the most important aspects of signal analysis, are developed for the first time in book form as special cases of the more general wave packets. This theory, initiated by Cordoba and Fefferman, has proved to be a valuable tool in operator theory and partial differential equations. Fine decompositions of signals by means of wave packets are also a new and promising tool in signal processing. The book begins with a refresher on basic wavelet and Gabor analysis, followed by the general theory of discrete and continuous wave packets. The presentation is distinguished by its focus on wave packets as objects worthy of independent study.
Containing an overview at the beginning of each chapter and many diverse examples, the book may be used as a text for a graduate course or seminar on modern mathematical methods in signal and image processing or applications of harmonic analysis to PDEs. Readers require only a background in real analysis and integration and a certain amount of mathematical maturity, with some familiarity with Gabor and wavelet systems being helpful but not mandatory. Researchers in pure and applied mathematics, physics, and engineering will also find the work to be an excellent reference. TOC:Preface. - Time-Frequency and Time-Scale Analyses. Fourier Transforms. Wavelet Transforms. Gabor and Short-time Fourier Transforms. Wigner Transform, Wigner Distribution, Rihaczek Transform. Curvelet Transform. Discretization. Other Examples of Transforms Analyzing the TFS Content (radar ambiguity function).- The Continuous Wave Packet Transform. Definition/Examples (Continuous Wavelet Transform/Windowed Fourier Transform). Reproducing Formulas. Approximate Plancherel Theorems. Changing the Parameters Defining a Surface.- Discrete Wave Packet Transform. Definition/Examples (Wavelets, Gabor Systems). Reproducing Systems. Necessary Conditions on the Analyzing Function. Sufficient Conditions on the Analyzing Function. Necessary Conditions on Parameter Set. Sufficient Conditions on Parameter Set. Characterizations.- Between Time-Frequency and Time-scale Analyses. $/Alpha$-Modulation Spaces. Flexible Gabor-Wavelet Decompositions. Atomic Decompositions and Banach Frames in Coorbit Spaces.- Bilinear Hilbert Transform. Introduction. Lacey-Thiele Proof of the Boundedness of BHT. A Wave Packet Proof of Carleson's Theorem.
Autor*in
Wojciech Czaja