Harmonic Analysis and Rational Approximation

✍ Scribed by Fournier J.-D., Grimm J. (eds.)

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 307
• Category: Library

✦ Table of Contents

front-matter.pdf
fulltext01.pdf
1 Introduction
2 The optical dispersion relations
3 Scattering of particles and complex energy
References
fulltext02.pdf
1 Analyticity and di.erentiability
1.1 Di.erentiability
1.2 Integrals
1.3 Power series expansions
1.4 Some properties of analytic functions
2 Analytic continuation and singularities
2.1 The problem of analytic continuation
2.2 Isolated singularities
2.3 Laurent expansion
2.4 Residue theorem
2.5 The logarithm
3 Continuation of a power series
4 Gevrey series
4.1 De.nitions
4.2 Exponential smallness and uniqueness
4.3 Gevrey summability
5 Borel summability
5.1 Connection with the usual Laplace transfor
5.2 Alien derivations
5.3 Real summation
References
fulltext03.pdf
1 Real and complex Fourier analysis
1.1 Fourier series
1.2 Fourier transforms
1.3 Harmonic and analytic functions
2 DFT, FFT, windows
3 The behaviour of f and  f
4 Wiener’s theorems
5 Laplace and Mellin transforms
5.1 Laplace
5.2 Mellin
References
fulltext04.pdf
1 Introduction
2 The Pad´e Table
3 Convergence
4 Examples
5 Calculation of Pad´e approximants
References
fulltext05.pdf
1 Classical Logarithmic Potential Theory
2 Polynomial Approximation of Analytic Functions
3 Approximation with Varying Weights — a background
4 Logarithmic Potentials with External Fields
5 Generalized Weierstrass Approximation Problem
6 Rational Approximation
References
fulltext06.pdf
1 Introduction
2 Orthogonal polynomials and Szeg¨o bases
2.1 Functions in the unit disc
2.2 Functions in the right half-plane
3 Wavelets
3.1 Orthonormal bases
3.2 Frames
References
fulltext07.pdf
1 Introduction
2 A short elementary probability theory refresher
3 Another proof of the CLT
4 Some extensions and related results
4.1 Other proofs of the CLT
4.2 Rate of convergence in the CLT
4.3 Other types of convergence
4.4 Bounds on the .uctuations
4.5 Brownian motion
4.6 Dependent random variables
5 Statistical Applications
6 Large deviations
7 Multifractal measures
References
fulltext08.pdf
1 Introduction
2 Real roots
3 Complex roots
3.1 Complex roots
3.2 Generalized monic polynomials
3.3 Strong disorder limit: classical homogeneous polynomial
3.4 Weak disorder limit: monic polynomials
3.5 Self-inversive polynomials
4 Conclusion
References
fulltext09.pdf
1 Introduction
2 Rational Interpolation
3 Convergence
4 Rational Interpolation with Noisy Data
5 Froissart Polynomial
6 Conclusions
References
fulltext10.pdf
1 Introduction
2 A Short View on the History
3 The Spectral Theory of Stationary Processes
3.1 Stationary Processes and Hilbert spaces
3.2 The Spectral Representation
3.3 The Isomorphism between Time Domain and Frequency domain. Linear Transformations of Stationary Processes
4 The Wold Decomposition and Forecasting
5 Rational Spectra, ARMA and State Space Systems
6 The Relation to System Identi.cation
References
fulltext11.pdf
1 Introduction
2 Parametric modeling for Power Spectral Density: ARMA and AR models
2.1 The Yule–Walker equations
3 AR and whitening process
3.1 Minimum phase .lter and stability
4 AR parameters estimation
5 The whitening .lter in the time domain
6 An example of whitening
References
fulltext12.pdf
1 Introduction
2 Linear time-invariant systems and their transfer functions
3 Function spaces and stability
3.1 Hardy spaces of the half-plane
3.2 Some notions of stability
4 Finite order LTI systems and their rational transfer functions
4.1 Controllability, observability and associated gramians
4.2 Hankel singular values and Hankel operator
5 Identi.cation and approximation
5.1 Hankel-norm approximation
5.2 L 2 -norm approximation
References
fulltext13.pdf
1 Introduction
2 Hardy spaces
3 Motivations from System Theory
3.1 Stochastic identi.cation
3.2 Harmonic Identi.cation
4 Some approximation problems
4.1 Analytic bounded extremal problems
4.2 Meromorphic and rational approximation
References
fulltext14.pdf
1 Introduction
2 Hamiltonian formalism
3 Integrable and nearly–integrable systems
3.1 The two–body problem
3.2 The three–body problem
4 Perturbation theory
4.1 Classical perturbation theory
4.2 KAM theory
5 A discrete model: the standard map
5.1 Link between periodic orbits and invariant curves
5.2 Perturbative series expansions
6 Numerical investigation of the break–down threshold
6.1 Pad´e approximants
6.2 Lyapunov’s method
6.3 Greene’s method
6.4 Results
References
fulltext15.pdf
1 Two elementary key examples – Basic concepts
1.1 Finite transition matrix and dynamical zeta function
1.2 Correlation functions and spectrum of the transfer operator
1.3 Basic concepts
2 Theorems of Ruelle, Keller, Pollicott, Dolgopyat...
References
Surveys and books
Analytical framework
Symbolic dynamics framework (H¨older-Lipschitz)
Di.erentiable framework
Numerics and aplications
fulltext16.pdf
1 An overview of the main properties of Turbulence
1.1 Qualitative Analysis of Turbulence
1.2 Statistical modeling of Eulerian turbulence
1.3 Vortex modeling for turbulence and oscillating singularities
2 Signal Processing Methods for Experiments on Turbulence
2.1 Some limitations of Fourier analysis
2.2 Multiresolution characterization and estimation of scaling law
2.3 Time-frequency methods for Lagrangian and Vorticity measurements
2.4 Mellin representation for self-similarity
References
fulltext17.pdf
1 Introduction
2 Interferometers
3 Servo systems
3.1 Introduction
3.2 Mathematical requirements on open loop transfer function
3.3 The Coulon’s solution
4 Conclusion and perspectives
References
back-matter.pdf