Density of Gabor Frames |
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Authors: | Ole Christensen Baiqiao Deng Christopher Heil |
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Institution: | Department of Mathematics, Technical University of Denmark, Building 303, 2800, Lyngby, Denmarkf1;Department of Mathematics, Columbus State University, Columbus, Georgia, 31907, , f2;School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, 30332, , f3 |
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Abstract: | A Gabor system is a set of time-frequency shifts S(g, Λ) ={e2 π ibxg(x − a)}(a, b) Λ of a function g L2(Rd). We prove that if a finite union of Gabor systems k = 1rS(gk, Λk) forms a frame for L2(Rd) then the lower and upper Beurling densities of Λ = k = 1r Λk satisfy D−(Λ) ≥ 1 and D + (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection k = 1r{gk(x − a)}a Γk of pure translates can form a frame for L2(Rd). |
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Keywords: | Beurling density frame frame of translates Gabor frame Riesz basis |
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