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DIRECTIONAL TIME–FREQUENCY ANALYSIS VIA CONTINUOUS FRAMES

Published online by Cambridge University Press:  30 April 2015

OLE CHRISTENSEN
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark email [email protected]
BRIGITTE FORSTER
Affiliation:
Fakultät für Informatik und Mathematik, Universität Passau, Innstr. 33, 94032 Passau, Germany email [email protected]
PETER MASSOPUST*
Affiliation:
Zentrum Mathematik, Lehrstuhl M6, Technische Universität München, Boltzmannstr. 3, 85747 Garching, Germany Helmholtz Zentrum München, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany email [email protected]
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Abstract

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Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in $L^{2}(\mathbb{R}^{n})$ based on Gabor systems in $L^{2}(\mathbb{R})$. The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for $L^{2}(\mathbb{R}),$ in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ${\it\epsilon}$-nets.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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