Pseudo-Differential Operators: Quantization and Signals : Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 19-24, 2006

by Feichtinger, Hans G.; Helffer, Bernard; Lamoureux, Michael P.; Lerner, Nicolas; Toft, Joachim

• ISBN13: 9783540682660
• ISBN10: 354068266X
• Format: Paperback
• Copyright: 2008-09-03
• Publisher: Springer Verlag

Summary

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hormander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century.

Table of Contents

Banach Gelfand Triples for Gabor Analysis	p. 1
Introduction	p. 1
Preliminaries	p. 3
Gabor Analysis on L[superscript 2]	p. 6
Time-Frequency Representations	p. 9
The Gelfand Triple (S[subscript 0], L[superscript 2], S[subscript 0]' (R[superscript d])	p. 12
The Spreading Function and Pseudo-Differential Operators	p. 19
Gabor Multipliers	p. 29
References	p. 31
Four Lectures in Semiclassical Analysis for Non Self-Adjoint Problems with Applications to Hydrodynamic Instability	p. 35
General Introduction	p. 35
Lecture 1: The Rayleigh-Taylor Model	p. 37
The Rayleigh-Taylor Model: Physical Origin	p. 37
Rayleigh-Taylor Mathematically	p. 40
Elementary Spectral Theory	p. 41
A Crash Course on h-Pseudodifferential Operators	p. 42
Application for Rayleigh-Taylor: Semi-Classical Analysis for K(h)	p. 44
Harmonic Approximation	p. 45
Instability of Rayleigh-Taylor: An Elementary Approach via WKB Constructions	p. 46
Lecture 2: Towards Non Self-Adjoint Models	p. 49
Instability for Kelvin-Helmholtz I: Physical Origin	p. 49
Around the [epsilon]-Pseudo-Spectrum	p. 50
Around the h-Family-Pseudospectrum	p. 51
The Davies Example by Hand	p. 52
Kelvin-Helmholtz II: Mathematical Analysis	p. 55
Other Toy Models	p. 58
Lecture 3: On Semi-Classical Subellipticity	p. 58
Introduction	p. 58
Non Subellipticity: Generic Result	p. 59
Link with the Standard Non-Hypoellipticity Results for Operators of Principal Type	p. 60
Elementary Proof for the Non-Subelliptic Model	p. 60
1/2 Semi-Classical Subellipticity	p. 62
Lecture 4: Other Non Self-Adjoint Models Coming from Hydrodynamics	p. 63
Introduction	p. 63
Quasi-Isobaric Model (Kull and Anisimov)	p. 65
Stationary Laminar Solution	p. 65
From the Physical Parameters to the Relevant Mathematical Parameters	p. 66
The Convection Velocity Model	p. 67
The Model for the Ablation Regime	p. 69
Semi-Classical Regimes for the Ablation Models	p. 71
Subellipticity II: At the Boundary of [Sigma](a[subscript 0])	p. 73
References	p. 75
An Introduction to Numerical Methods of Pseudodifferential Operators	p. 79
Signal Processing and Pseudodifferential Operators	p. 79
Introduction to Seismic Imaging	p. 79
Introduction to Pseudodifferential Operators	p. 82
A Jump in Dimension	p. 87
Boundedness of the Operators	p. 89
Manipulating Pseudodifferential Operators	p. 93
Composition of Operators	p. 93
Asymptotic Series	p. 95
Oscillatory Integrals	p. 96
Other Pseudo-Topics	p. 99
Numerical Implementations	p. 100
Sampling and Quantization Error in Signal Processing	p. 100
The Discrete Fourier Transform and Periodization Errors	p. 102
Direct Numerical Implementation via the DFT	p. 103
Operations Count	p. 106
Numerical Implementation via Product-Convolution Operators	p. 107
Almost Diagonalization via Wavelet and Gabor Bases	p. 108
Gabor Multipliers	p. 110
Short Time Fourier Transforms and Their Multipliers	p. 110
Gabor Transforms and Gabor Multipliers	p. 113
Gabor Transforms in Practice	p. 116
Sampled Space	p. 116
Sampling in the Frequency Domain	p. 119
Partitions of Unity and Frequency Subsampling	p. 121
Uniform POUs	p. 126
Seismic Imaging	p. 130
Wavefield Extrapolation	p. 130
References	p. 132
Some Facts About the Wick Calculus	p. 135
Elementary Fourier Analysis via Wave Packets	p. 135
The Fourier Transform of Gaussian Functions	p. 135
Wave Packets and the Poisson Summation Formula	p. 136
Toeplitz Operators	p. 140
On the Weyl Calculus of Pseudodifferential Operators	p. 141
A Few Classical Facts	p. 141
Symplectic Invariance	p. 143
Composition Formulas	p. 145
Definition and First Properties of the Wick Quantization	p. 147
Definitions	p. 147
The Garding Inequality with Gain of One Derivative	p. 151
Variations	p. 152
Energy Estimates via the Wick Quantization	p. 156
Subelliptic Operators Satisfying Condition (P)	p. 156
Polynomial Behaviour of Some Functions	p. 158
Energy Identities	p. 162
The Fefferman-Phong Inequality	p. 164
The Semi-Classical Inequality	p. 164
The Sjostrand Algebra	p. 165
Composition Formulas	p. 166
Sketching the Proof	p. 167
A Final Comment	p. 172
Appendix	p. 172
Cotlar's Lemma	p. 172
References	p. 173
Schatten Properties for Pseudo-Differential Operators on Modulation Spaces	p. 175
Introduction	p. 175
Preliminaries	p. 178
Schatten-Von Neumann Classes for Operators Acting on Hilbert Spaces	p. 185
Schatten-Von Neumann Classes for Operators Acting on Modulation Spaces	p. 188
Continuity and Schatten-Von Neumann Properties for Pseudo-Differential Operators	p. 191
References	p. 201
List of Participants	p. 203
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