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Transactions of the American Mathematical Society

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On accumulated spectrograms


Authors: Luís Daniel Abreu, Karlheinz Gröchenig and José Luis Romero
Journal: Trans. Amer. Math. Soc. 368 (2016), 3629-3649
MSC (2010): Primary 81S30, 45P05, 94A12, 42C25, 42C40
DOI: https://doi.org/10.1090/tran/6517
Published electronically: April 8, 2015
MathSciNet review: 3451888
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Abstract: We study the eigenvalues and eigenfunctions of the time-frequency localization operator $ H_\Omega $ on a domain $ \Omega $ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $ \Omega \subseteq \mathbb{R}^{2d}$. Indeed, in analogy to the classical theory of Landau-Pollak-Slepian, the number of eigenvalues of $ H_\Omega $ in $ [1-\delta , 1]$ is equal to the measure of $ \Omega $ up to an error term depending on the perimeter of the boundary of $ \Omega $. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition of unity of the given domain $ \Omega $. We derive asymptotic, non-asymptotic, and weak-$ L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $ \Omega $ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.


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Additional Information

Luís Daniel Abreu
Affiliation: Acoustics Research Institute, Austrian Academy of Science, Wohllebengasse 12-14 A-1040, Vienna, Austria
Email: daniel@mat.uc.pt

Karlheinz Gröchenig
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Email: karlheinz.groechenig@univie.ac.at

José Luis Romero
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Email: jose.luis.romero@univie.ac.at

DOI: https://doi.org/10.1090/tran/6517
Received by editor(s): April 30, 2014
Received by editor(s) in revised form: June 16, 2014
Published electronically: April 8, 2015
Additional Notes: The first author was supported by the Austrian Science Fund (FWF) START-project FLAME (“Frames and Linear Operators for Acoustical Modeling and Parameter Estimation”) 551-N13
The second author was supported in part by the project P26273-N25 and the National Research Network S106 SISE of the Austrian Science Fund (FWF)
The third author gratefully acknowledges support by the project M1586-N25 of the Austrian Science Fund (FWF)
Article copyright: © Copyright 2015 American Mathematical Society

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