A conjecture on the frame bounds of the multidimensional Gabor transform with complex-valued envelopes

Summary

The discrete Gabor functions used for the n-D Gabor transform with constant complex-valued envelopes of the Gabor functions are assumed to form a frame. The approximate analytic expressions for the frame bounds of this frame are conjectured. These expressions can be used to analytically study both the discretization error of the continuous Gabor transform and the stability of the discrete Gabor transform in dependence on the regular but generally oblique lattice of the central points of the Gabor functions in the 2n-D phase space. The algorithm of constructing the optimum phase-space lattice of the central points of the Gabor functions with constant complex-valued envelopes is then conjectured. The algorithm of constructing the phase-space lattice of the central points of the Gabor functions with slightly varying complex-valued envelopes is also conjectured.

Keywords

Gabor function, Gabor transform, frame bounds, discretization error, phase space, metric tensor.

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