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Characterization of L. Schwartz’ convolutor and multiplier spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\) by the short-time Fourier transform

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Abstract

A new definition of the short-time Fourier transform for temperate distributions is presented and its mapping properties are investigated. K.-H. Gröchenig and G. Zimmermann characterized the spaces \(\mathcal S \) and \(\mathcal S '\) of rapidly decreasing functions and temperate distributions, respectively, by their short-time Fourier transform. Following an idea of G. Zimmermann, we give analogous characterizations of the spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\). These spaces, being (PLB)-spaces, have a much more complicated structure than \(\mathcal S \) and \(\mathcal S '\), which is the reason why we have to use the technical machinery of L. Schwartz’ theory of vector-valued distributions.

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Acknowledgments

We are indebted to Prof. G. Zimmermann who directed the second author’s attention to the mapping properties of the short-time Fourier transform and to their inversion. Preliminary versions of Propositions 4, 4., 5, 2., 7, 2., 10, 1., 2., and 11, 1., 2 are due to him.

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  1. Institut für Mathematik, Universität Innsbruck, Technikerstraße 19a, 6020,  Innsbruck, Austria

    Christian Bargetz & Norbert Ortner

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  1. Christian Bargetz
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  2. Norbert Ortner
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Correspondence to Christian Bargetz.

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Bargetz, C., Ortner, N. Characterization of L. Schwartz’ convolutor and multiplier spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\) by the short-time Fourier transform. RACSAM 108, 833–847 (2014). https://doi.org/10.1007/s13398-013-0144-4

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  • DOI: https://doi.org/10.1007/s13398-013-0144-4

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