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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Coorbit spaces and wavelet coefficient decay over general dilation groups
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by Hartmut Führ PDF
Trans. Amer. Math. Soc. 367 (2015), 7373-7401 Request permission

Abstract:

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on $\textrm {L}^2(\mathbb {R}^d)$. 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 $\textrm {L}^{p}$-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
  • Received by editor(s): August 13, 2012
  • Received by editor(s) in revised form: January 9, 2014
  • Published electronically: October 3, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 7373-7401
  • MSC (2010): Primary 22D10, 42C15, 46E35, 42C40
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06376-9
  • MathSciNet review: 3378833