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