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Analyticity and causality in physics of the XIX and XXth centuries;
resonances in atomic physics and physics of particles;
dispersion relations in optical physics and physics of particles;
rays profiles;
S matrix; Regge's poles; Re-Summations (Padé,...).
G. Turchetti, Analyticity and resonances in physics
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Singularities of analytic functions of the complex variable;
What to do with divergent series or with analytic functions singularities?
B. Candelpergher, Complex analysis, a reminder (old and new)
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Use of resummations in physics of particles and field theory.
Mathematics around Padé approximants; Padé table; convergence theorem
;
function approximation; use and interpretation precaution;
numerical algorithms.
E.B. Saff, Theory and practice of Padé table
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Real and complex Fourier analysis; DFT, FFT, windows;
links between real and complex behaviours of a function and that of
real and complex behaviours of its Fourier transform; Wiener's theorems;
Laplace and Mellin transformss.
J.R. Partington, Fourier transform and complex analysis
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Reminders about polynomial and rational approximation (Tchebychev);
introduction to Hardy spaces of the complex plane unit disk;
approximation in Hardy classes and links with extrapolation (Carleman);
meromorphic and rational approximation.
E.B. Saff, Analytic and rational approximation
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Convergence in probability theory; pivot role of (various versions of)
central limit theorem in probability and statistics; estimation;
some modern ideas and works; speed of convergence, large deviations,
functional techniques.
P. Collet, Statistics and probability, around the central limit theorem
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(a) Reminder of hamiltonian mecanics; celestial mecanics;
developments of perturbations.
L. Biasco, Dynamical systems and celestial mecanics.
A reminder on hamiltonian dynamic. Three bodies problem.
(b) Coupling constant developments; series in mechanics,
celestial in particular. Approach by Padé resummations.
Study of the analyticity domain and comparison between various methods.
A. Celletti, Dynamical systems and celestial mecanics.
Series in celestial mecanics; analyticity domains.
(c) Theorems of Ruelle, Keller, Pollicott, Dolgopyat.
ii. Résonances de systčmes hamiltoniens.
V. Baladi, Dynamical systems and celestial mecanics.
Resonances in hyperbolic and hamiltonian systems
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(a)
Analyticity and causality, boundedness and staility in
signal theory and control theory. Stationnary processes in time and
frequency domains, linear transforms of stationnary processes,
Wold decomposition and prediction, filtrin, AR, ARMA models, and
systems internal representation.
M. Deistler, Control and signal processing. Historical and a reminder
(b) Spectral theory, Fourier analysis, Hankel operators;
inverse spectral problem and balanced realizations of transfer functions.
N. Nikolski, Control and signal processing. Spectral theory and control theory
(c)Identification and function theory: rational and analytic
approximation of transfer functions from frequencies measurements,
bounded extremal problems; aymptotic behaviour of poles and errors
related to spectral analysis of Hankel and Toeplitz operators
and potential theory.
L. Baratchart, Control and signal processing. Recent approaches
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(a) Froissart phenomena.
J.-D. Fournier, Rational approximation with noise. In real or parameters space
(b) Szegö analysis; noise identification.
B. Dujardin, Rational approximation with noise. In Fourier space
(c) Random spectral measure; spectral decomposition of stationnary gaussian
processes; obtention of rational approximants for stationnary gaussian
processes using functional analysis arguments.
Rational approximation with noise.
A. Gombani, Rational approximation with noise.
Spectral decomposition of stationnary gaussian processes
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Problems of prediction and detection of interferometric
gravitational waves; captors noises and their characteristics;
non gaussianity; uses of rational approximations.
J.-Y. Vinet, F. Bondu, E. Cuoco,
Interferometric gravitational waves detection
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From signals treatment point of view. Use of Mellin transform.
P. Borgnat, Turbulence and information
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Applying pure mathematics / purifying applied mathematics.
Interactions between information theory and complex analysis.
New open issues.
E.B. Saff, Round--table course: disguises of
a concept in various fields
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Fourier series and integrals from 1807 to 2003.
J.-P. Kahane, The Fourier transform in mathematics history
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Multiple series, Hartogs phenomenon, singularities in several variables;
extension problems, Erhenpreis fundamental principle; rational functions,
rational convexity and approximation problems; extension to several variables
of results on polynomial speed of approximation (Bernstein like theorem).
N. Sibony, Singularities and approximation in
several complex variables
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From Fourier to wavelets, by orthogonal polynomials and Szegö bases
J.R. Partington, Good bases
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A constrained approximation problem for reflexive Banach spaces, with
applications. Martin Smith, York University
Abstract: (This is joint work with Isabelle Chalendar and Jonathan
Partington) I will formulate a general approximation problem involving
reflexive, smooth Banach spaces and provide an explicit solution. Two
applications will be given. I will first give a solution to a Bounded
Completion Problem, which involves approximation via Hardy class
functions. I will also discuss the construction of minimal vectors and
invariant subspaces for a linear operator.
Zolotarev's problem: asymptotics and applications to numerical
linear algebra.
Bernhard Beckermann
Laboratoire de Mathematiques Appliquees (ANO) CNRS FRE 2222
Universite des Sciences et Technologies de Lille, France
Abstract:
Given two disjoint compact sets E and F in the complex plane,
Zolotarov's problem consists of finding a rational function of
degree n being of modulus at most one on E, and as large as
possible on F. Asymptotics for such optimal rational functions are
known, e.g., in the case of real intervals. Here we consider the case
of discrete sets, e.g., F being the set of 2n=th roots of unity.
The main tool in these considerations is some extremal problem
in logarithmic potential theory, namely the equilibrium for some
condenser with maximum charge constrains.
F. Seyfert, Some tapas of approximation theory - applications towards other fields
of mathematics