Time-Frequency and Wavelet Analysis of Functions with Symmetry Properties

By (author) Holger Rauhut

Extent: 220, 220 pages

Publisher: Logos Verlag Berlin

Subjects: Technical Sciences: Computer Science, Engineering, IT, & Math

Language: English

(Paperback) (Published)

(January 2005)

ISBN: 9783832507787

5.71 x 8.27 inches

Price: $61.00

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Synopsis
Contributing Authors

Die Dissertation behandelt die Zeit-Frequenz- und Wavelet-Analysis von Funktionen auf dem Euklidischen Raum R d , die invariant unter gewissen Untergruppen der orthogonalen Gruppe O(d) sind. Ein Hauptaugenmerk liegt dabei auf den radialsymmetrischen Funktionen; die Theorie wird aber in voller Allgemeinheit entwickelt. Dabei wird ein abstrakter Zugang uber quadratintegrierbare Darstellungen lokalkompakter Gruppen gewahlt, da hierbei sowohl Zeit-Frequenz-Analyse and Wavelet-Analyse gleichzeitig behandelt werden konnen. Nach Einfuhrung wichtiger Grundlagen, etwa der harmonischen Analysis radialer Funktionen, Hypergruppen und quadratintegrablen Darstellungen, wird die Theorie der abstrakten kontinuierlichen Wavelet-Transformation invarianter Vektoren bzw. Funktionen entwickelt. Auf die Beispiele der gefensterten Fourier-Transformation und der kontinuierlichen Wavelet-Transformation radialer Funktionen wird im Detail eingegangen. Der zweite Teil des Buches befasst sich mit Diskretisierungen der kontinuierlichen Transformation. Mittels des Konzepts einer Multiresolutionsanalysis gelingt es im Spezialfall der radialen Wavelet-Analysis im R 3, eine orthonormale radiale Wavelet-Basis fur den Raum L 2 rad ( R 3 ) aller quadratintegrierbaren radialen Funktionen zu konstruieren. Um das allgemeine Diskretisierungsproblem zu losen wird die sogenannte Feichtinger-Grochenig-Theorie bzw. Koorbitraumtheorie verallgemeinert. Dies erlaubt die Konstruktion von Banach-Frames und atomaren Zerlegungen fur sogenannte Koorbitraume invarianter Vektoren bzw. Funktionen. Im Fall der Wavelet-Analysis sind dies Besov und Triebel-Lizorkin-Raume und fur den Fall der Zeit-Frequenz-Analyse Modulationsraume. Die Konstruktion radialer Wavelet-Frames und radialer Gabor-Frames wird im Detail durchgefuhrt. Dieser Teil des Buches ist auch als Einfuhrung in die Koorbitraumtheorie geeignet, die bislang nur in Form von Forschungsartikeln vorliegt.

By (author) Holger Rauhut

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