Kernel theorems for modulation spaces
For researchers in harmonic analysis and signal processing, this provides a complete kernel characterization for modulation spaces, revealing advantages over distribution theory.
The paper fully characterizes the continuity of linear operators on modulation spaces M^p for all 1 ≤ p ≤ ∞ by the membership of their kernel to mixed modulation spaces, establishing an equivalence of operator norm and kernel norm. This extends Feichtinger's kernel theorem and has no counterpart in distribution theory.
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 $1\leq p\leq\infty$, by the membership of its kernel to (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 temperate distributions, our results do not have a couterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism upon 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 sated in the same spirit.