Geometric Space–Frequency Analysis on Manifolds

Abstract

This paper gives a survey of methods for the construction of space–frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace–Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley–Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley–Wiener frames in \(L_2(\mathbf {M})\), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in \(L_2(\mathbf {M})\), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on \(\mathbf {M}\). For compact Riemannian manifolds the theory extends to \(L_p\) and associated Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called product property for eigenfunctions of certain operators and proves cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay. The general theory is exemplified with the help of various concrete and relevant examples which include the unit sphere and the Poincaré half plane.

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Acknowledgments

The authors would like to thank the ESI (Erwin Schrödinger Institute, University Vienna), where the joint work for this paper has begun after the workshop on Time-Frequency Analysis (Spring 2014) and CIRM (Centre international de recontres mathematique, Luminy, Marseille), where the three authors had the chance to continue their work on this manuscript during the period of Hans Feichtinger’s Morlet Chair (winter term 2014/2015). The third author was supported in part by the National Geospatial-Intelligence Agency University Research Initiative (NURI), Grant HM1582-08-1-0019.

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Correspondence to Hartmut Führ.

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Communicated by Stephan Dahlke.

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Feichtinger, H.G., Führ, H. & Pesenson, I.Z. Geometric Space–Frequency Analysis on Manifolds. J Fourier Anal Appl 22, 1294–1355 (2016). https://doi.org/10.1007/s00041-015-9457-3

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Keywords

  • Compact Homogeneous Manifolds
  • Parseval Frame
  • Cubature Formulas
  • Besov Spaces
  • Boundary Geometry