Abstract
It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions. In this paper, we clarify exactly how “bad” such Gabor expansions are, we make it clear precisely where the edge is between “enough” and “too little,” and we find a remedy for their shortcomings in terms of a certain summability method. This is done through an investigation of somewhat more general sequences of points in the time-frequency plane than lattices (all of Nyquist density), which in a sense yields information about the uncertainty principle on a finer scale than allowed by traditional density considerations. An important role is played by certain Hilbert scales of function spaces, most notably by what we call the Schwartz scale and the Bargmann scale, and the intrinsically interesting fact that the Bargmann transform provides a bounded invertible mapping between these two scales. This permits us to turn the problems into interpolation problems in spaces of entire functions, which we are able to treat.
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Communicated by A.J.E.M. Janssen
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Lyubarskii, Y.I., Seip, K. Convergence and summability of Gabor expansions at the Nyquist density. The Journal of Fourier Analysis and Applications 5, 127–157 (1999). https://doi.org/10.1007/BF01261606
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DOI: https://doi.org/10.1007/BF01261606