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Directional time-frequency analysis via continuous frames.
Bull. Austral. Math. Soc. 92, 268-281 (2015)
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 (Formula presented.) based on Gabor systems in (Formula presented.). 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 (Formula presented.) 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 (Formula presented.)-nets.
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Publication type Article: Journal article
Document type Scientific Article
Keywords Complex B-spline ; Directionally Sensitive Time–frequency Decomposition ; Discrete And Continuous Frames ; Discretisation Of The Sphere ; Ridge Function; Representations; Dimension
ISSN (print) / ISBN 0004-9727
Quellenangaben Volume: 92, Issue: 2, Pages: 268-281
Publisher Cambridge Univ. Press
Publishing Place Cambridge
Institute(s) Institute of Computational Biology (ICB)