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An Inverse Problem for Localization Operators
A classical result of time-frequency analysis, obtained by I. Daubechies in
1988, states that the eigenfunctions of a time-frequency localization operator
with circular localization domain and Gaussian analysis window are the Hermite
functions. In this contribution, a converse of Daubechies' theorem is proved.
More precisely, it is shown that, for simply connected localization domains, if
one of the eigenfunctions of a time-frequency localization operator with
Gaussian window is a Hermite function, then its localization domain is a disc.
The general problem of obtaining, from some knowledge of its eigenfunctions,
information about the symbol of a time-frequency localization operator, is
denoted as the inverse problem, and the problem studied by Daubechies as the
direct problem of time-frequency analysis. Here, we also solve the
corresponding problem for wavelet localization, providing the inverse problem
analogue of the direct problem studied by Daubechies and Paul.Comment: 18 pages, 1 figur
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