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Coorbit spaces and wavelet coefficient decay over general dilation groups


Author: Hartmut Führ
Journal: Trans. Amer. Math. Soc. 367 (2015), 7373-7401
MSC (2010): Primary 22D10, 42C15, 46E35, 42C40
Published electronically: October 3, 2014
MathSciNet review: 3378833
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Abstract: We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on $ {\rm L}^2(\mathbb{R}^d)$. It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with.

We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed $ {\rm L}^{p}$-estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions.

We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.


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Additional Information

Hartmut Führ
Affiliation: Lehrstuhl C für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
Email: fuehr@matha.rwth-aachen.de

DOI: https://doi.org/10.1090/S0002-9947-2014-06376-9
Keywords: Square-integrable group representation, continuous wavelet transform, coorbit spaces, Banach frames, vanishing moments, shearlets, temperately embedded orbits
Received by editor(s): August 13, 2012
Received by editor(s) in revised form: January 9, 2014
Published electronically: October 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society