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Smooth projections and the construction of smooth Parseval frames of shearlets

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Abstract

Smooth orthogonal projections with good localization properties were originally studied in the wavelet literature as a way to both understand and generalize the construction of smooth wavelet bases on \(L^{2}(\mathbb {R})\). Smoothness plays a critical role in the construction of wavelet bases and their generalizations as it is instrumental to achieve excellent approximation properties. In this paper, we extend the construction of smooth orthogonal projections to higher dimensions, a challenging problem in general for which relatively few results are found in the literature. Our investigation is motivated by the study of multidimensional nonseparable multiscale systems such as shearlets. Using our new class of smooth orthogonal projections, we construct new smooth Parseval frames of shearlets in \(L^{2}(\mathbb {R}^{2})\) and \(L^{2}(\mathbb {R}^{3})\).

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Authors and Affiliations

  1. University of Houston, Houston, TX, 77004, USA

    Bernhard G. Bodmann, Demetrio Labate & Basanta R. Pahari

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  1. Bernhard G. Bodmann
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  2. Demetrio Labate
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  3. Basanta R. Pahari
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Correspondence to Demetrio Labate.

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Communicated by: Gitta Kutyniok

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The research in this paper was partially supported by NSF DMS-1720452.

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Bodmann, B.G., Labate, D. & Pahari, B.R. Smooth projections and the construction of smooth Parseval frames of shearlets. Adv Comput Math 45, 3241–3264 (2019). https://doi.org/10.1007/s10444-019-09736-3

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  • DOI: https://doi.org/10.1007/s10444-019-09736-3

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