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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On Toeplitz operators and localization operators
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by Luís Daniel Abreu and Nelson Faustino PDF
Proc. Amer. Math. Soc. 143 (2015), 4317-4323 Request permission


This note is a contribution to a problem of Lewis Coburn concerning the relation between Toeplitz operators and Gabor-Daubechies localization operators. We will show that, for any localization operator with a general window $w\in \mathcal {F}_{2}({\mathbb {C}})$ (the Fock space of analytic functions square-integrable on the complex plane), there exists a differential operator of infinite order $D$, with constant coefficients explicitly determined by $w,$ such that the localization operator with symbol $f$ coincides with the Toeplitz operator with symbol $Df$. This extends results of Coburn, Lo and Engliš, who obtained similar results in the case where $w$ is a polynomial window. Our technique of proof combines their methods with a direct sum decomposition in true polyanalytic Fock spaces. Thus, polyanalytic functions are used as a tool to prove a theorem about analytic functions.
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Additional Information
  • Luís Daniel Abreu
  • Affiliation: Acoustic Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040, Vienna, Austria – and – CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal
  • Email:
  • Nelson Faustino
  • Affiliation: CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal
  • Address at time of publication: Departamento de Matemática Aplicada, IMECC-Unicamp, CEP 13083-859, Campinas, SP, Brasil
  • Email:
  • Received by editor(s): December 26, 2012
  • Received by editor(s) in revised form: March 20, 2013, and August 7, 2013
  • Published electronically: June 16, 2015
  • Additional Notes: Both authors were supported by CMUC and FCT (Portugal), through European program COMPETE/FEDER and by FCT project PTDC/MAT/114394/2009. The first author was also supported by Austrian Science Foundation (FWF) project “Frames and Harmonic Analysis” and START-project FLAME (“Frames and Linear Operators for Acoustical Modeling and Parameter Estimation”, Y 551-N13). The second author was also supported by São Paulo Research Foundation (FAPESP) through the grant 13/07590-8.
  • Communicated by: Pamela B. Gorkin
  • © Copyright 2015 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 4317-4323
  • MSC (2010): Primary 47B32, 30H20; Secondary 81R30, 81S30
  • DOI:
  • MathSciNet review: 3373930