%0 Journal Article
%T Asymptotic Boundary Forms for Tight Gabor Frames and Lattice Localization Domains
%A H. G. Feichtinger
%A K. Nowak
%A M. Pap
%J Journal of Applied Mathematics and Physics
%P 1316-1342
%@ 2327-4379
%D 2015
%I Scientific Research Publishing
%R 10.4236/jamp.2015.310160
%X We consider Gabor localization operators <img alt=\"\" src=\"Edit_be0b57b3-15b0-4f4e-b677-cce356e54eee.jpg\" />&nbsp;defined by two parameters, the generating function <img alt=\"\" src=\"Edit_f1c11bb0-dccf-4e0e-96de-9d3125465735.bmp\" />&nbsp;of a tight Gabor frame <img alt=\"\" src=\"Edit_515a018a-d140-4df0-b71e-ead804c8fe86.jpg\" />, indexed by a lattice <img alt=\"\" src=\"Edit_065d30af-cf03-4dfc-8999-e12d93f7841c.jpg\" />, and a domain <img alt=\"\" src=\"Edit_f2690d7f-f244-49d6-8536-1dd660bd75ca.jpg\" />&nbsp;whose boundary consists of line segments connecting certain points of . We provide an explicit formula for the boundary form <img alt=\"\" src=\"Edit_1dc92091-7027-4748-89d4-cba09affd655.jpg\" />, the normalized limit of the projection functional <img alt=\"\" src=\"Edit_496f026b-db1f-430c-8edf-70bbfb730a77.jpg\" />, where <img alt=\"\" src=\"Edit_837f6c82-56e8-472a-9229-92bba729b271.jpg\" />&nbsp;are the eigenvalues of the localization operators <img alt=\"\" src=\"Edit_6bdc21de-b283-4d75-ba5e-f1679dcde5bc.jpg\" />&nbsp;applied to dilated domains <img alt=\"\" src=\"Edit_413390e8-3bb3-47b2-bf39-44666642744a.jpg\" />, R is an integer and is<img alt=\"\" src=\"Edit_712d4e38-75bc-48cc-b402-b2e1fde7a94b.jpg\" /> the area of the fundamental domain. The boundary form expresses quantitatively the asymptotic interactions between the generating function <img alt=\"\" src=\"Edit_36e7ee79-abfd-41bd-8739-71cc3dddfd13.bmp\" />&nbsp;and the oriented boundary <img alt=\"\" src=\"Edit_ea5240dd-6865-4ab2-8f17-c4a22845c225.jpg\" />&nbsp;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 <img alt=\"\" src=\"Edit_429d1a28-85c6-4f92-a2f2-48cedd2d0f2d.bmp\" />&nbsp;bounded above corresponds to controlling the relative dimensionality of the localization problem.
%K Toeplitz Operators
%K Phase Space Localization
%K Tight Gabor Frames
%K Semi-Classical Limit
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=60745