Generalized Functions and Fourier Analysis : Dedicated to Stevan Pilipović on the Occasion of His 65th Birthday

This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC. Contents 6 Preface 8 On Temperate Distributions Decaying at Infinity 10 1. Introduction 10 2. Preliminaries 11 2.1. Subspaces of temperate distributions 11 2.2. SG-symbols and operators 12 3. Temperate distributions decaying at infinity and mapping properties 15 3.1. Rapidly decreasing distributions 15 3.2. s-decreasing distributions, sεR 21 Acknowledgement 26 References 26 Transport in a Stochastic Goupillaud Medium 28 1. Introduction 28 2. Setting up the Goupillaud medium 30 2.1. Dyadic deterministic structure 31 2.2. The stochastic model 32 3. Limits as the time step goes to zero 33 3.1. A convergence result for càdlàg functions 33 3.2. Convergence of characteristic curves 35 3.3. Convergence of approximate solutions 36 4. Conclusion 38 Acknowledgement 38 References 38 Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces 40 0. Introduction 40 1. Preliminaries 42 1.1. The Pilipović spaces 42 1.2. Spaces of Hermite series expansions 43 2. Embedding properties for quasi-Banach spaces contained in Pilipović distribution spaces 44 References 52 Blow-up Phenomena for Solutions of Discrete Nonlinear p-Laplacian Parabolic Equations on Networks 54 1. Introduction 54 2. Preliminaries 55 3. Blow-up for the case p=2 59 4. Blow-up for the general case 61 5. Blow-up of the Fujita type 63 References 66 Generalized Function Algebras Containing Spaces of Periodic Ultradistributions 68 1. Introduction 68 2. Spaces of periodic ultradifferentiable functions and their duals 70 3. Structure theorem for periodic ultradistributions 74 4. Impossibility result on the multiplication of periodic ultradistributions 76 5. Algebras of periodic generalized functions of class {Mp} and {Mp} 78 5.1. Definition and basic properties 78 5.2. Projective description of G{Mp} 2π 80 5.3. Generalized point values 82 6. Embedding of periodic ultradistributions 83 7. Regular periodic generalized functions of class {Mp} and {Mp} 84 References 86 On General Prime Number Theorems with Remainder 88 1. Introduction 88 2. Preliminaries and notation 90 2.1. Beurling generalized number systems 90 2.2. Fourier transforms and distributions 91 3. A Tauberian theorem with remainder 91 4. The PNT with Nyman’s remainder 94 5. A Cesàro version of the PNT with remainder 99 Acknowledgement 102 References 102 Inverse Function Theorems for Generalized Smooth Functions 104 1. Introduction 104 1.1. Basic notions 106 2. Local inverse function theorems 112 3. Global inverse function theorems 116 4. Conclusions 120 Acknowledgement 120 References 120 The Stochastic LQR Optimal Control with Fractional Brownian Motion 124 1. Introduction 124 2. Theoretical background 127 2.1. The SLQR problem: existence of solution 127 2.1.1. Inhomogeneous deterministic LQR problem. 128 2.1.2. Strong and mild solutions. 129 2.2. Fractional Brownian motion 129 2.3. White noise analysis and chaos expansions 130 2.3.1. Gaussian white noise space. 131 2.3.2. Stochastic processes. 133 2.3.3. Operators. 135 2.3.4. Stochastic integration and Wick multiplication. 136 2.3.5. The fractional transform operatorM(H). 139 2.3.6. Fractional Gaussian white noise space. 140 3. The Stochastic LQR problem with fractional Brownian motion 141 3.1. The fractional operator M 141 3.1.1. Fractional integral. 144 3.2. The optimal control problem 145 3.3. Further extensions 154 4. An example involving operators from Malliavin calculus 155 Acknowledgement 157 References 157 Multi-soliton Collision for Essentially Nonintegrable Equations 161 1. Introduction 161 2. Weak asymptotics method: the main idea 165 3. Multi-soliton interaction 168 3.1. General construction 168 3.2. Analysis of the model equations 171 4. Non-uniqueness phenomenon 173 Acknowledgement 176 References 177 Microlocal Solvability and Subellipticity of Several Classes of Pseudodifferential Operators with Involutive Characteristics 179 1. Introduction and formulation of the main results 179 2. Proof of Propositions 3–5 187 References 191 An Observation of the Subspaces of S' 193 1. Introduction 193 2. Proof of Theorem 1.1 194 2.1. A reduction and preliminaries 194 2.2. The kernel of R 195 2.3. The surjectivity of R: X→XV 196 2.4. The openness of R 196 3. Applications 198 3.1. Schwartz space 198 3.2. The space S'm 199 3.3. The Hasumi space S'e 199 3.4. The space D' 199 Acknowledgement 199 References 200 Ultradifferentiable Functions of Class Mt,σp and Microlocal Regularity 201 1. Introduction 201 1.1. Notation 203 1.2. Ultradifferentiable functions and wave-front sets 203 2. Mt,σp sequences and the corresponding regularity classes 205 2.1. The defining sequence Mt,σp 205 2.2. Regularity classes Et,σp 206 3. Local analysis of distributions with respect to εt,σ 209 4. Wave-front sets with respect to εt,σ 213 4.1. Pseudolocal property of WFt,σ 216 4.2. Intersections and unions of WFt,σ and the corresponding singular supports 217 Acknowledgment 219 References 219 Matrix Parameterized Pseudo-differential Calculi on Modulation Spaces 222 0. Introduction 222 1. Preliminaries 223 1.1. An extended family of pseudo-differential calculi 224 1.2. Modulation spaces 227 1.3. Schatten–von Neumann classes 229 2. Algebraic and continuity properties 231 2.1. Composition properties 238 3. An idea of quantization 239 Acknowledgement 240 References 240 An Application of Internal Objects to Microlocal Analysis in Generalized Function Algebras 243 1. Introduction 243 2. Internal objects 244 3. Principles from nonstandard analysis 246 4. Internal subsets of *Rd and internal functions on *Rd 247 5. Moderateness and M∞-regularity 249 6. M∞-microlocal regularity 251 7. Consistency with M∞-regularity 253 8. Connection with G∞-microlocal regularity 255 Acknowledgment 256 References 256 Rotation Invariant Ultradistributions 258 1. Introduction 258 2. Preliminaries and auxiliary results 259 2.1. Spherical harmonics 259 2.2. Ultradistributions 260 2.3. Ultradistributions on Sn-1 and spherical harmonics 261 2.4. Ultradistributions on R × Sn-1 262 3. Spherical representations of ultradistributions 263 4. Rotation invariant ultradistributions 268 Acknowledgement 271 References 271 Eigenvalue Problems of Toeplitz Operators in Bargmann–Fock Spaces 273 1. Introduction 273 2. Bargmann transform and Bargmann–Fock space 273 2.1. Kernel function of Bargmann transform 273 2.2. Bargmann transform B 274 2.3. The Bargmann–Fock space BF(Cn) 274 3. Orthogonal projection and Toeplitz operators 275 3.1. Orthogonal projection 275 3.2. Toeplitz operators 276 3.3. Eigenvalue problems of Toeplitz operators on Bargmann–Fock spaces 276 4. Gabor transform and resolution of identity 277 4.1. Gabor transform 277 4.2. Resolution of identity (unity) 277 5. Daubechies localization operator 277 5.1. Daubechies operator in a Bargmann–Fock space 278 5.2. Eigenvalue problem of the Daubechies operator 279 References 279 Front Matter....Pages i-viii On Temperate Distributions Decaying at Infinity....Pages 1-18 Transport in a Stochastic Goupillaud Medium....Pages 19-30 Hilbert Space Embeddings for Gelfand–Shilov and Pilipović Spaces....Pages 31-44 Blow-up Phenomena for Solutions of Discrete Nonlinear p-Laplacian Parabolic Equations on Networks....Pages 45-58 Generalized Function Algebras Containing Spaces of Periodic Ultradistributions....Pages 59-78 On General Prime Number Theorems with Remainder....Pages 79-94 Inverse Function Theorems for Generalized Smooth Functions....Pages 95-114 The Stochastic LQR Optimal Control with Fractional Brownian Motion....Pages 115-151 Multi-soliton Collision for Essentially Nonintegrable Equations....Pages 153-170 Microlocal Solvability and Subellipticity of Several Classes of Pseudodifferential Operators with Involutive Characteristics....Pages 171-184 An Observation of the Subspaces of \( \mathcal{S}^{\prime} \) ....Pages 185-192 Ultradifferentiable Functions of Class \( M_p^{\tau ,\sigma } \) and Microlocal Regularity....Pages 193-213 Matrix Parameterized Pseudo-differential Calculi on Modulation Spaces....Pages 215-235 An Application of Internal Objects to Microlocal Analysis in Generalized Function Algebras....Pages 237-251 Rotation Invariant Ultradistributions....Pages 253-267 Eigenvalue Problems of Toeplitz Operators in Bargmann–Fock Spaces....Pages 269-276