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DIRECTIONAL TIME–FREQUENCY ANALYSIS VIA CONTINUOUS FRAMES

Part of: Numerical approximation and computational geometry Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  30 April 2015

OLE CHRISTENSEN ,
BRIGITTE FORSTER  and
PETER MASSOPUST
OLE CHRISTENSEN
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark email ochr@dtu.dk
BRIGITTE FORSTER
Affiliation:
Fakultät für Informatik und Mathematik, Universität Passau, Innstr. 33, 94032 Passau, Germany email brigitte.forster@uni-passau.de
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 massopust@ma.tum.de
*
✉massopust@ma.tum.de
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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.

Keywords

discrete and continuous framesridge functiondirectionally sensitive time–frequency decompositioncomplex B-splinediscretisation of the sphere

MSC classification

Primary: 42C15: General harmonic expansions, frames
Secondary: 42C40: Wavelets and other special systems 65D07: Splines
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 268 - 281
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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