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Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

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Abstract

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence.

In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝd by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe.

The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.

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Authors and Affiliations

  1. Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Wojciech Czaja

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Wojciech Czaja

  3. Department of Statistics, Stanford University, Stanford, CA, 94305, USA

    Gitta Kutyniok

  4. Department of Mathematics and Computer Science, Saint Louis University, 221 North Grand Blvd., St. Louis, MO, 63103, USA

    Darrin Speegle

Authors
  1. Wojciech Czaja
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  2. Gitta Kutyniok
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  3. Darrin Speegle
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Correspondence to Wojciech Czaja.

Additional information

Communicated by Karlheinz Gröchenig.

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Czaja, W., Kutyniok, G. & Speegle, D. Beurling Dimension of Gabor Pseudoframes for Affine Subspaces. J Fourier Anal Appl 14, 514–537 (2008). https://doi.org/10.1007/s00041-008-9026-0

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  • DOI: https://doi.org/10.1007/s00041-008-9026-0

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