On time-frequency analysis and pseudo-differential operators for vector-valued functions

This thesis treats different aspects of time-frequency analysis and pseudodifferential operators, with particular emphasis on techniques involving vector-valued functions and operator-valued symbols. The vector (Banach) space is either motivated by an application as in Paper I, where it is a space of stochastic variables, or is part of a general problem as in Paper II, or arises naturally from problems for scalar-valued operators and function spaces, as in Paper V. Paper III and IV fall outside this framework and treats algebraic aspects of time-frequency analysis and pseudodifferential operators for scalar-valued symbols and functions that are members of modulation spaces. Paper IV builds upon Paper III and applies the results to a filtering problem for second-order stochastic processes. Paper I treats the Wigner distribution of a Gaussian weakly harmonizable stochastic process defned on Rd. Paper II extends recent continuity results for pseudodifferential and localization operators, with symbols in modulation spaces, to the vector/operator-framework, where the vector space is a Hilbert or a Banach space. In Paper III we give algebraic results for the Weyl product acting on modulation spaces. We give suffcient conditions for a weighted modulation space to be an algebra under theWeyl product, and we also give necessary conditions for unweighted modulation spaces. In Paper IV we discretize the results of Paper III by means of a Gabor frame delined by a Gaussian function. Finally, Paper V deals with pseudodifferential operators with symbols that are almost periodic in the first variable. We show that such operators may be transformed to Fourier multiplier operators with operator- valued symbols such that the transformation preserves positivity and operator composition