Asymptotic boundary forms for tight Gabor frames and lattice localization domains

Feichtinger, H. G. and Nowak, K. and Pap, Margit (2015) Asymptotic boundary forms for tight Gabor frames and lattice localization domains. JOURNAL OF APPLIED MATHEMATICS AND PHYSICS, 3 (10). pp. 1316-1342. ISSN 2327-4352

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Abstract

We consider Gabor localization operators Gφ,Ω defined by two parameters, the generating function φ of a tight Gabor frame {φλ}λ∈Λ , parametrized by the elements of a given lattice Λ ⊂ R 2 , i.e. a discrete cocompact subgroup of R 2 , and a lattice localization domain Ω ⊂ R 2 with its boundary consisting of line segments connecting points of Λ. We find an explicit formula for the boundary form BF(φ, Ω) = AΛ limR→∞ P F (Gφ,RΩ) R , the normalized limit of the projection functional P F(Gφ,Ω) = P∞ i=0 λi(Gφ,Ω)(1 − λi(Gφ,Ω)), where λi(Gφ,Ω) are the eigenvalues of the localization operators Gφ,Ω applied to dilated domains RΩ, R is an integer and AΛ is the area of the fundamental domain of the lattice Λ. Although the lattice Λ is also a parameter of the localization operator we assume that it is fixed and we do not list it explicitly in our notation. The boundary form expresses quantitatively the asymptotic interactions between the generating function φ of a tight Gabor frame and the oriented boundary ∂Ω of a lattice localization domain 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. It provides an evaluation framework for finding the best asymptotic matching between pairs consisting of generating functions φ and lattice domains Ω. It takes into account directional information involving outer normal vectors of the linear segments constituting the boundary of Ω and the weighted sums over the corresponding half spaces of the absolute value squares of the reproducing kernels obtained out of φ. In the context of tight Gabor frames and Gabor localization domains placing an upper bound on the square of the L 2 norm of the generating function φ divided by the area of the fundamental domain of lattice Λ corresponds to keeping the relative redundancy of the frame bounded. Keeping the area of the localization domain Ω bounded above corresponds to controlling the relative dimensionality of the localization problem.

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