Preprint revised TIME FREQUENCY ANALYSIS OF PSEUDODIFFERENTIAL OPERATORS

In this paper we apply a time frequency approach to the study of pseudodi er ential operators Both the Weyl and the Kohn Nirenberg correspondences are considered In order to quantify the time frequency content of a function or distribution we use certain function spaces called modulation spaces We deduce a time frequency characterization of the twisted product of two symbols and and we show that modulation spaces provide the natural setting to exactly control the time frequency content of from the time frequency content of and As a consequence we discuss some boundedness and spectral properties of the corresponding operator with symbol Introduction A pseudodi erential operator can be de ned through the Weyl or the Kohn Nirenberg correspondence by bijectively assigning to any distributional symbol S R n a linear operator T S R n S R so that the properties of the operator are in an appropriate way re ected in the properties of the symbol One way to construct a pseudodi erential operator is as a superposition of time frequency shifts Even though this is a classical idea going back to H Weyl this interpretation has reblossomed in recent years in the context of the study of the harmonic analysis in the Heisenberg group As a consequence a number of methods from the time frequency analysis have been employed to the study of pseudodi erential operators for instance In this paper we are interested in pseudodi erential operators whose symbols satisfy certain integrability conditions in the time frequency plane and are not necessarily smooth The interest of these classes of operators stems partly from electrical engineering appli cations in particular signal processing and time varying ltering theory where operators arising from the Weyl correspondence are used as models for time frequency or time varying lters cf for instance In this context the symbol is interpreted as Mathematics Subject Classi cation Primary S G Secondary C B