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 ≤ p ≤ ∞, 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
Kernel Theorems for Modulation Spaces / Cordero, Elena; Nicola, Fabio. - In: JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS. - ISSN 1069-5869. - STAMPA. - 25:1(2019), pp. 131-144.
|Titolo:||Kernel Theorems for Modulation Spaces|
|Data di pubblicazione:||2019|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/s00041-017-9573-3|
|Appare nelle tipologie:||1.1 Articolo in rivista|
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