Abstract
The shearlet representation has gained increasingly more prominence in recent years as a flexible and efficient mathematical framework for the analysis of anisotropic phenomena. This is achieved by combining traditional multiscale analysis with a superior ability to handle directional information. In this paper, we introduce a class of shearlet smoothness spaces which is derived from the theory of decomposition spaces recently developed by L. Borup and M. Nielsen. The introduction of these spaces is motivated by recent results in image processing showing the advantage of using smoothness spaces associated with directional multiscale representations for the design and performance analysis of improved image restoration algorithms. In particular, we examine the relationship of the shearlet smoothness spaces with respect to Besov spaces, curvelet-type decomposition spaces and shearlet coorbit spaces. With respect to the theory of shearlet coorbit space, the construction of shearlet smoothness spaces presented in this paper does not require the use of a group structure.
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Notes
Here the \(\mathcal{Q}\)-moderate function w is the first axis projection, and the sequence of points x i ∈Q i is given by x i =(2j,0).
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Acknowledgements
D.L. and P.N. were partially supported by NSF grants DMS 1008900 and DMS 1005799. This work was initiated while D.L. was visiting the NuHAG in Vienna. A part of this work was performed while L.M. was visiting the Department of Mathematics at the University of Houston.
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Communicated by Stephan Dahlke.
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Labate, D., Mantovani, L. & Negi, P. Shearlet Smoothness Spaces. J Fourier Anal Appl 19, 577–611 (2013). https://doi.org/10.1007/s00041-013-9261-x
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DOI: https://doi.org/10.1007/s00041-013-9261-x