Asymptotic boundary forms for tight Gabor frames and lattice localization domains

Abstract

We consider Gabor localization operators $G_{\ddot{I}•,\hat{I}\text{c◯}}$ defined by two parameters, the generating function $\ddot{I}•$ of a tight Gabor frame $\{\ddot{I}{•}_{\hat{I}}»{\}}_{\hat{I}»\in \hat{I}›}$, parametrized by the elements of a given lattice $\hat{I}›\subset {\mathbb{R}}^2$, i.e. a discrete cocompact subgroup of ${\mathbb{R}}^2$, and a lattice localization domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ with its boundary consisting of line segments connecting points of $\hat{I}›$. We find an explicit formula for the boundary form $BF(\ddot{I}•,\hat{I}\text{c◯})={\text{A}}_{\hat{I}}›{\lim }_{R\to \infty }\frac{PF(G_{\ddot{I}•,R\hat{I}\text{c◯}})}{R}$, the normalized limit of the projection functional $PF(G_{\ddot{I}•,\hat{I}\text{c◯}})={\sum }_{i=0}^{\infty }\hat{I}{»}_i(G_{\ddot{I}•,\hat{I}\text{c◯}})(1-\hat{I}{»}_i(G_{\ddot{I}•,\hat{I}\text{c◯}}))$, where $\hat{I}{»}_i(G_{\ddot{I}•,\hat{I}\text{c◯}})$ are the eigenvalues of the localization operators $G_{\ddot{I}•,\hat{I}\text{c◯}}$ applied to dilated domains $R\hat{I}\text{c◯}$, $R$ is an integer and ${\text{A}}_{\hat{I}}›$ is the area of the fundamental domain of the lattice $\hat{I}›$.

35 pages