Density and duality theorems for regular Gabor frames

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• 作者：Mads Sielemann Jakobsen ^{msja@dtu.dk" class="auth_mail" title="E-mail the corresponding author} ; Jakob Lemvig ; ^{jakle@dtu.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 42C15 ; secondary ; 42C40 ; 43A25 ; 43A32 ; 43A70
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：270
• 期：1
• 页码：229-263
• 全文大小：611 K

文摘

We investigate Gabor frames on locally compact abelian groups with time–frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the “size” of the subgroup. From these density results we are able to extend the classical Wexler–Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time–frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.