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

  • Christian BargetzEmail author
  • Norbert Ortner
Original Paper


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.


Temperate and (very) rapidly decreasing distributions and functions Fourier transform Short-time Fourier transform 

Mathematics Subject Classification (2010)

Primary 42B10 Secondary 46F12 



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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Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Institut für MathematikUniversität Innsbruck InnsbruckAustria

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