Beurling Dimension of Gabor Pseudoframes for Affine Subspaces |
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Authors: | Wojciech Czaja Gitta Kutyniok Darrin Speegle |
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Institution: | (1) Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland;(2) Department of Mathematics, University of Maryland, College Park, MD 20742, USA;(3) Department of Statistics, Stanford University, Stanford, CA 94305, USA;(4) Department of Mathematics and Computer Science, Saint Louis University, 221 North Grand Blvd., St. Louis, MO 63103, USA |
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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 ℝ
d
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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Keywords: | Beurling density Beurling dimension Frame Gabor system Discrete Hausdorff dimension Homogeneous Approximation Property Mass dimensions Nyquist density Pseudoframe Pseudoframe for subspaces |
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