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Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions. (English) Zbl 1351.42039
The authors study frame bounds using a Gaussian window function and solve a conjecture on the form of the underlying lattice to find optimal lower and upper bounds for the frame constants. The main result is:
Theorem 2.1. Consider the window function \(g_0(t)=2^{1/4}\exp{(-\pi t^2)}\). Among all separable lattices with \((\alpha\beta)^{-1}\in\mathbb N\) fixed, the square lattice maximizes \(A\) and minimizes \(B\).
The following play a role:
1.
Gabor system \({\mathcal G}(g,\Lambda)\) for \(L^2(\mathbb R^d)\), generated by a fixed non-zero window function \(g\in L^2(\mathbb R^d)\), using an index set \(\Lambda\in\mathbb R^{2d}\) and time frequency shift \(\lambda=(x,\omega)\in\mathbb R^d\times\mathbb R^d\) \[ \pi(\lambda)g(t)=M_{\omega}T_xg(t)=e^{2\pi i\omega\cdot t}g(t-\lambda),\;x,\omega,\lambda,t\in\mathbb R^d. \]
2.
The system \({\mathcal G}(g,\Lambda)\) is a frame if it satisfies the inequalities \[ A\| f\|_2^2\leq \sum_{\lambda\in\Lambda}\,|\langle f,\pi (\lambda)g\rangle|^2\leq B\| f\|_2^2,\;\forall f\in L^2(\mathbb R^d). \]
3.
The index set \(\Lambda\in\mathbb R^{2d}\) is a lattice if it is genrated by an invertible \(2d\times 2d\) matrix \(S\).
4.
The lattice is separable if \(S\) can take the form \[ S=\begin{pmatrix} \alpha I & 0 \\ 0 & \beta I\end{pmatrix}. \]
5.
Fine estimates of the Jacobi theta functions \[ \theta_3(s)=\sum_{k=-\infty}^{\infty}e^{-\pi k^2s}, \theta_4(s)=\sum_{k=-\infty}^{\infty}\,(-1)^ke^{-\pi k^2s}. \]

MSC:
42C15 General harmonic expansions, frames
06B75 Generalizations of lattices
14H42 Theta functions and curves; Schottky problem
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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