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Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 131–144 | Cite as

Kernel Theorems for Modulation Spaces

  • Elena CorderoEmail author
  • Fabio Nicola
Article

Abstract

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces \(M^p\) for every \(1\le p\le \infty \), by the membership of its kernel in (mixed) modulation spaces. Whereas Feichtinger’s kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz’ kernel theorem for tempered distributions, our results do not have a counterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism over distribution theory, as already emphasized in Feichtinger’s manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on \(M^{p,q}\) a similar characterization is not expected, but sufficient conditions for boundedness can be stated in the same spirit.

Keywords

Time-frequency analysis Gabor frames Modulation spaces 

Mathematics Subject Classification

42B35 42C15 47G30 81Q20 

Notes

Acknowledgements

This research was partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), Italy.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly
  2. 2.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly

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