H. G. Feichtinger^1*, K. Nowak², M. Pap³

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¹ Faculty of Mathematics, University Vienna, Wien, Austria.
² Department of Computer Science, Drexel University, Philadelphia, PA, USA.
³ Faculty of Sciences, University of Pécs, Pécs, Hungary.
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Abstract: We consider Gabor localization operators  defined by two parameters, the generating function  of a tight Gabor frame , indexed by a lattice , and a domain  whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form , the normalized limit of the projection functional , where  are the eigenvalues of the localization operators  applied to dilated domains , R is an integer and is the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function  and the oriented boundary  from the point of view of the projection functional, which measures to what degree a given trace class operator fails to be an orthogonal projection. Keeping the area of the localization domain  bounded above corresponds to controlling the relative dimensionality of the localization problem.

Keywords: Toeplitz Operators, Phase Space Localization, Tight Gabor Frames, Semi-Classical Limit

Cite this paper: Feichtinger, H. , Nowak, K. and Pap, M. (2015) Asymptotic Boundary Forms for Tight Gabor Frames and Lattice Localization Domains. Journal of Applied Mathematics and Physics, 3, 1316-1342. doi: 10.4236/jamp.2015.310160.

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