Time-frequency analysis: function spaces and applications (Q2902319)

This paper is a survey on some recent results on the modulation spaces introduced by \textit{H. G. Feichtinger} [``Modulation spaces on locally compact abelian groups'', Technical Report, University Vienna (1983); reprinted in: M. Krishna, R. Radha, S. Thangavelu (eds.), ``Wavelets and Their Applications'', Proc. Internat. Conf., Chennai, India, January 2002. New Delhi: Allied Publishers, 99--140 (2003), \url{http://www.univie.ac.at/nuhag-php/bibtex/open_files/fe031_modspa03.pdf}], and their applications in the study of the boundedness of the localization operators and certain classes of Fourier integral operators (FIO). The authors give some basic elements of time-frequency analysis, explaining the meaning of the localization operators as filters in signal theory (Section 2), and then define and study the properties of modulation spaces (Sections 3 and 4). General necessary and sufficient conditions for boundedness and Schatten classes of localization operators are formulated in Section 5. In Section 6 an almost diagonalization result with respect to Gabor frames is proposed for some classes of FIO. In this way, the continuity of these FIOs in appropriate modulation spaces with weight \(\mu\in M_{V_s}\), \(s\geq 0\), \(V_s= (1+|x|^2+|\omega|^2)^{s/2}\) is proved.