Search results for "modulation spaces"
showing 2 items of 2 documents
Annihilating sets for the short time Fourier transform
2010
Abstract We obtain a class of subsets of R 2 d such that the support of the short time Fourier transform (STFT) of a signal f ∈ L 2 ( R d ) with respect to a window g ∈ L 2 ( R d ) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L 2 -norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.
On Fourier integral operators with Hölder-continuous phase
2018
We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a H\"older-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the H\"older exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a H\"older singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions and relevant counterexamples. The proofs rely on techniques from Time-frequency Analysis.