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On Time-Frequency Analysis and Pseudo-Differential Operators for Vector-Valued Functions On time-frequency analysis and pseudo-differential operators for vector-valued functions Acta Wexionensia No 145/2008 Mathematics On time-frequency analysis and pseudo-differential operators for vector-valued functions Patrik Wahlberg Växjö University Press On time-frequency analysis and pseudo-differential operators for vector- valued functions. Thesis for the degree of Doctor of Philosophy, Växjö Univer- sity, Sweden 2008. Series editor: Kerstin Brodén ISSN: 1404-4307 ISBN: 978-91-7636-612-7 Printed by: Intellecta Docusys, Göteborg 2008 Abstract This thesis treats di®erent aspects of time-frequency analysis and pseudodi®erential operators, with particular emphasis on tech- niques involving vector-valued functions and operator-valued sym- bols. The vector (Banach) space is either motivated by an appli- cation 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 pseudodi®erential operators for scalar-valued symbols and functions that are mem- bers of modulation spaces. Paper IV builds upon Paper III and applies the results to a ¯ltering problem for second-order stochastic processes. Paper I treats the Wigner distribution of a Gaussian weakly harmonizable stochastic process de¯ned on Rd. Paper II extends recent continuity results for pseudodi®erential and localization op- erators, 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 su±cient conditions for a weighted modulation space to be an algebra under the Weyl 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 de¯ned by a Gaussian function. Finally, Paper V deals with pseudodi®erential operators with symbols that are al- most periodic in the ¯rst 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. v Acknowledgments I thank my supervisor and friend Joachim Toft for accepting me as a Ph.D. student and for the stimulating collaboration on Paper III of the thesis. I am grateful for his suggestion of the problem, which originates in one of his papers, and for his cru- cial contribution to the paper. Also Paper IV originates from our collaboration, and in fact we worked out the essential calculations together in VÄaxjÄoin January 2006 while working on Paper III. I would also like to express my gratitude to my mathemat- ics teachers in Lund: Alexandru Aleman, Adrian Constantin, Nils Dencker, Sigmundur Gudmundsson, Henrik Kalisch, Anders Melin, Arne Meurman and Nils Nilsson, as well as my teachers in Vienna: Hans Feichtinger and Karlheinz GrÄochenig. They have all been excellent and inspiring teachers of graduate and seminar courses. Moreover, I have enjoyed nice discussions with Mikael Signahl on many mathematical topics. Finally I thank Per{Anders Svensson for helping me with typographical questions. vi Contents Abstract v Acknowledgments vi Introduction 1 Bibliography 16 I The Wigner distribution of Gaussian weakly harmonizable stochastic processes 19 Abstract 21 1 Introduction 21 2 Gaussian stochastic processes on Rd 22 3 Scalar-valued weakly harmonizable stochastic processes 23 4 Spectral representation of a weakly harmonizable process 26 5 The Wigner distribution of a deterministic distribution 28 6 The Wigner distribution of a weakly harmonizable Gaussian stochas- tic process 29 7 The Wigner distribution of stationary Gaussian processes 31 Bibliography 35 II Vector-valued modulation spaces and localization operators with operator-valued symbols 37 Abstract 39 1 Introduction 39 1.1 De¯nitions and notation . 42 2 The STFT of vector-valued tempered distributions 44 p;q d 3 Modulation spaces Mm (R ;B) 47 3.1 Retract property, admissible windows, Wiener amalgam space characterization, properties . 47 p;q d 3.2 The dual of Mm (R ;B) when B is reflexive . 51 3.3 Weakly stationary Hilbert space valued tempered distributions 56 3.4 Gabor frame theory for Hilbert space valued modulation spaces 56 4 Localization operators and Weyl operators 60 4.1 Operators on Banach space valued function spaces with sym- bols in Lp;q ............................ 60 4.2 Operators on Hilbert space valued function spaces with sym- bols in M 1 ............................ 62 Bibliography 67 III Weyl product algebras and modulation spaces 71 Abstract 73 1 Introduction 73 2 Preliminaries 76 3 Continuity of the Weyl product on modulation spaces 82 4 Necessary and su±cient conditions for Weyl product algebras 92 5 Some remarks on the Wiener property 99 vii Bibliography 100 IV Gabor discretization of the Weyl product for modulation spaces and ¯ltering of nonstationary stochastic processes 103 Abstract 105 1 Introduction 105 2 Notation and preliminaries 108 3 The Weyl correspondence, modulation spaces and Gabor expansions110 3.1 The Weyl quantization and the Weyl product . 110 3.2 Modulation spaces, the Weyl product and pseudodi®erential operators . 111 3.3 Gabor frames for modulation spaces de¯ned by a Gaussian . 113 4 Gabor frame discretization of the Weyl product 116 4.1 Polynomial o®-diagonal decay of the matrices . 120 4.2 Exponential o®-diagonal decay of the matrices . 122 5 Stochastic processes and generalized stochastic processes 124 6 Filtering and the equation for the optimal ¯lter 125 7 The optimal ¯lter 128 8 Approximation of the global error 131 9 Approximation by Gabor multipliers 134 Bibliography 137 V A transformation of almost periodic pseudodi®erential op- erators to Fourier multiplier operators with operator-valued symbols 141 Abstract 143 1 Introduction 143 2 Notation and preliminaries 145 3 Almost periodic functions and pseudodi®erential operators 147 4 A transformation of symbols for a.p. pseudodi®erential operators 154 Bibliography 164 viii Introduction In the thesis we will present results on time-frequency analysis and pseu- dodi®erential operators. Three of the ¯ve papers in the thesis concerns the framework of vector-valued functions and operator-valued symbols, where vector space means a Hilbert space or a Banach space. Two papers treat algebraic questions for pseudodi®erential operators with scalar-valued sym- bols and functions, that are members of modulation spaces. As we hope to show, the extension to vector-valued functions and operator- valued symbols may be done for several reasons. In the ¯rst place, as in Paper I and II, it is sometimes of interest in appli- cations of time-frequency analysis and pseudodi®erential calculus to study the vector/operator-valued extension. In Paper I we discuss the Wigner distribution of certain Gaussian stochastic processes, which are considered Hilbert space valued functions on Rd. In Paper II we generalize some recent continuity results for pseudodi®erential operators with modulation space symbols to the vector/operator-valued situation. Secondly, as in Paper V, a problem in the theory of scalar-valued functions and symbols may lead to problems in the vector/operator-valued frame- work. More precisely, Paper V treats pseudodi®erential operators whose symbols are almost periodic in the ¯rst variable. Introduction of Fourier series techniques transforms such operators to Fourier multiplier operators with operator-valued symbols. As a background to the ¯ve papers in the thesis, this introduction is intended to give a brief survey of time-frequency analysis and pseudodi®er- ential operators in the scalar-valued setup, necessarily incomplete due to the great scope and diversity of the subject. The account aims in particular at the relatively recent algebraic and continuity results for the nonsmooth sym- bol classes consisting of modulation spaces, that are essential background for Papers II, III and IV. Time-frequency analysis is an interdisciplinary area of research, with branches in pure and applied mathematics, physics and signal theory. It has one root in the work by Wigner and Weyl on the mathematical founda- tions of quantum mechanics from the 1930s [10,11]. Wigner tried to create a probability density function for a particle in the phase (position-momentum) space by introduction of the Wigner distribution Z ¡d=2 ¡ih»;¿i Wf;f (x; ») = (2¼) f(x + ¿=2)f(x ¡ ¿=2)e d¿ Rd associated to a function f de¯ned on Rd. It gives a description of f over the phase space (x; ») 2 R2d, which in the quantum mechanical interpretation may be seen as a probability density function for a particle in phase space. However, the Wigner distribution is nonnegative only for a very particular class of functions, { Gaussians, according to Hudson's theorem, so the prob- ability density interpretation is dubious. This phenomenon is one aspect of the Heisenberg Uncertainty Principle, which says that a particle cannot be localized in small regions in the phase space. In fact, small regions in phase 3 Introduction space may not be assigned probability, so the fact that the Wigner distribu- tion does not serve as a probability density is not surprising. Nevertheless, if the phase space resolution is decreased by convolving the Wigner distribu- tion with a su±ciently wide Gaussian function, a nonnegative phase space distribution is obtained, which is consistent with the Uncertainty Principle due to the decrease of resolution. In mathematics, Uncertainty Principles occur in several forms in har- monic analysis. One basic version is the inequality µZ ¶1=2 µZ ¶1=2 2 2 2 2 1 2 (t ¡ a) jf(t)j ) dt (» ¡ b) jF f(»)j ) d» ¸ kfkL2 R R 2 for all a; b 2 R and f 2 L2(R), which is a rather immediate consequence of the Cauchy{Schwarz inequality and the Plancherel theorem.
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