Date: 13 Sep 2013

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.