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Asymptotic Boundary Forms for Tight Gabor Frames and Lattice Localization Domains

DOI: 10.4236/jamp.2015.310160, PP. 1316-1342

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

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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.

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