Abstract
We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.
Highlights
1 Introduction and Main Result The moving spirit of this work originates in a conjecture formulated by Strohmer and Beaver [27]. They claim that the condition number of the Gabor frame operator for a Gaussian window and a hexagonal lattice of fixed density δ > 1 is minimal among all lattice of same fixed density δ
We show that, starting from a rectangular lattice, the upper frame bound is always lowered by shearing the lattice, or, equivalently, by chirping the window
The function Fg−γ takes its maximum whenever (x, ω) ∈ Z × Z. This implies that in this case the optimal upper frame bound for a Gabor frame with standard Gaussian window is given by the formula
Summary
A Gabor system for the Hilbert space L2(R) is the set of time-frequency shifted versions of a window function g ∈ L2(R) with respect to some index set ⊂ R2. Due to the work of Lyubarskii [23] and Seip [26] we know that for a Gaussian window any lattice of density δ > 1 generates a Gabor frame for L2(R). In this case we know that we cannot have a frame for δ = 1 because of the Balian-Low theorem [3,22]. This implies that we cannot obtain an orthonormal basis with timefrequency shifted Gaussian windows which makes it interesting to study sharp frame bounds for the Gaussian. Within this setting we are interested in the tightest possible bounds and will show that the hexagonal lattice minimizes the upper frame bound
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