Journal of Fourier Analysis and Applications

, Volume 14, Issue 4, pp 514–537

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Article

DOI: 10.1007/s00041-008-9026-0

Cite this article as:
Czaja, W., Kutyniok, G. & Speegle, D. J Fourier Anal Appl (2008) 14: 514. doi:10.1007/s00041-008-9026-0

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.

Keywords

Beurling densityBeurling dimensionFrameGabor systemDiscrete Hausdorff dimensionHomogeneous Approximation PropertyMass dimensionsNyquist densityPseudoframePseudoframe for subspaces

Mathematics Subject Classification (2000)

42C1542C40

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  • Wojciech Czaja
    • 1
    • 2
  • Gitta Kutyniok
    • 3
  • Darrin Speegle
    • 4
  1. 1.Institute of MathematicsUniversity of WrocławWrocławPoland
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of StatisticsStanford UniversityStanfordUSA
  4. 4.Department of Mathematics and Computer ScienceSaint Louis UniversitySt. LouisUSA