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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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