Abstract.
We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an Lp,q space and M∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.
Similar content being viewed by others
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wahlberg, P. Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols. Integr. equ. oper. theory 59, 99–128 (2007). https://doi.org/10.1007/s00020-007-1504-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-007-1504-2