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We study sharp frame bounds of Gabor frames with the standard Gaussian windowand prove that the square lattice optimizes both the lower and the upper framebound among all rectangular lattices. This proves a conjecture of Floch, Alard& Berrou (as reformulated by Strohmer & Beaver). The proof is based on refinedlog-convexity/concavity estimates for the Jacobi theta functions $\theta_3$ and$\theta_4$.

Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions