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arXiv · 1202.5841

An Inverse Problem for Localization Operators

Luis Daniel Abreu·Monika Doerfler
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Abstract

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.

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Luis Daniel Abreu, Monika Doerfler. 2012-07-23. An Inverse Problem for Localization Operators. https://doi.org/10.1088/0266-5611%2F28%2F11%2F115001

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