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

**Ludek Klimes**
### 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 2*n*-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.

### Whole paper

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In: Seismic Waves in Complex 3-D Structures, Report 18,
pp. 109-114, Dep. Geophys., Charles Univ., Prague, 2008.

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