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Pseudo-Differential Operators Quantization and Signals

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ISBN-10: 354068266X

ISBN-13: 9783540682660

Edition: 2008

Authors: Hans G. Feichtinger, Bernard Helffer, Michael P. Lamoureux, Nicolas Lerner, Joachim Toft

List price: $69.95
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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.
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Book details

List price: $69.95
Copyright year: 2008
Publisher: Springer Berlin / Heidelberg
Publication date: 8/11/2008
Binding: Paperback
Pages: 214
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 0.792
Language: English

Bernard Helffer is a Professor in the Department of Mathematics at Universite Paris-Sud. He has published more than 200 papers in mathematics and mathematical physics and authored five books. In 2011 he was awarded the Prix de l'Etat by the French Academy of Sciences.

Banach Gelfand Triples for Gabor Analysis
Gabor Analysis on L[superscript 2]
Time-Frequency Representations
The Gelfand Triple (S[subscript 0], L[superscript 2], S[subscript 0]' (R[superscript d])
The Spreading Function and Pseudo-Differential Operators
Gabor Multipliers
Four Lectures in Semiclassical Analysis for Non Self-Adjoint Problems with Applications to Hydrodynamic Instability
General Introduction
Lecture 1: The Rayleigh-Taylor Model
The Rayleigh-Taylor Model: Physical Origin
Rayleigh-Taylor Mathematically
Elementary Spectral Theory
A Crash Course on h-Pseudodifferential Operators
Application for Rayleigh-Taylor: Semi-Classical Analysis for K(h)
Harmonic Approximation
Instability of Rayleigh-Taylor: An Elementary Approach via WKB Constructions
Lecture 2: Towards Non Self-Adjoint Models
Instability for Kelvin-Helmholtz I: Physical Origin
Around the [epsilon]-Pseudo-Spectrum
Around the h-Family-Pseudospectrum
The Davies Example by Hand
Kelvin-Helmholtz II: Mathematical Analysis
Other Toy Models
Lecture 3: On Semi-Classical Subellipticity
Non Subellipticity: Generic Result
Link with the Standard Non-Hypoellipticity Results for Operators of Principal Type
Elementary Proof for the Non-Subelliptic Model
1/2 Semi-Classical Subellipticity
Lecture 4: Other Non Self-Adjoint Models Coming from Hydrodynamics
Quasi-Isobaric Model (Kull and Anisimov)
Stationary Laminar Solution
From the Physical Parameters to the Relevant Mathematical Parameters
The Convection Velocity Model
The Model for the Ablation Regime
Semi-Classical Regimes for the Ablation Models
Subellipticity II: At the Boundary of [Sigma](a[subscript 0])
An Introduction to Numerical Methods of Pseudodifferential Operators
Signal Processing and Pseudodifferential Operators
Introduction to Seismic Imaging
Introduction to Pseudodifferential Operators
A Jump in Dimension
Boundedness of the Operators
Manipulating Pseudodifferential Operators
Composition of Operators
Asymptotic Series
Oscillatory Integrals
Other Pseudo-Topics
Numerical Implementations
Sampling and Quantization Error in Signal Processing
The Discrete Fourier Transform and Periodization Errors
Direct Numerical Implementation via the DFT
Operations Count
Numerical Implementation via Product-Convolution Operators
Almost Diagonalization via Wavelet and Gabor Bases
Gabor Multipliers
Short Time Fourier Transforms and Their Multipliers
Gabor Transforms and Gabor Multipliers
Gabor Transforms in Practice
Sampled Space
Sampling in the Frequency Domain
Partitions of Unity and Frequency Subsampling
Uniform POUs
Seismic Imaging
Wavefield Extrapolation
Some Facts About the Wick Calculus
Elementary Fourier Analysis via Wave Packets
The Fourier Transform of Gaussian Functions
Wave Packets and the Poisson Summation Formula
Toeplitz Operators
On the Weyl Calculus of Pseudodifferential Operators
A Few Classical Facts
Symplectic Invariance
Composition Formulas
Definition and First Properties of the Wick Quantization
The Garding Inequality with Gain of One Derivative
Energy Estimates via the Wick Quantization
Subelliptic Operators Satisfying Condition (P)
Polynomial Behaviour of Some Functions
Energy Identities
The Fefferman-Phong Inequality
The Semi-Classical Inequality
The Sjostrand Algebra
Composition Formulas
Sketching the Proof
A Final Comment
Cotlar's Lemma
Schatten Properties for Pseudo-Differential Operators on Modulation Spaces
Schatten-Von Neumann Classes for Operators Acting on Hilbert Spaces
Schatten-Von Neumann Classes for Operators Acting on Modulation Spaces
Continuity and Schatten-Von Neumann Properties for Pseudo-Differential Operators
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