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Padé and Symmetrized Padé Approximants in Data
Processing
Jean-Daniel Fournier (1) (2) member of the
CNRS fournier_at_obs-nice.fr
Bénédicte Dujardin (1) (2) (3) (4) Ph.D.
student dujardin_at_obs-nice.fr
Talk presented by J.-D. Fournier at the GWDAW
2004
(1) member of ARTEMIS, a department of the
Observatoire de la Côte d Azur , affiliated
with the CNRS
(2) postal address Observatoire de Nice, BP
4229, 06304 NICE Cedex 4, FRANCE
(3) expected date of defense spring 2005
(4) teaching assistant at the Université de
Nice Sophia Antipolis
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Padé and Symmetrized Padé Approximants in Data
Processing
1. Analytic continuation and Fourier tools 2.
Natural spectra, resonances or worse 
rationality and quasi rationality 3. Rational
Approximation  AR, Szegö, ARMA,
interpolation, Hermite Padé, MPPA 4. The
energy spectrum  Taylor series at the origin.
The outer and inner part of the spectrum 5.
The Padé approximant of the spectrum.
Alternative definition  the symmetrized Padé
approximant 6. Success and pitfall 7. Test on
simulated data and real data from a Virgo E.Run
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Analytic Continuation and Fourier Tools I.
Fourier Transform
Inverse Fourier Transform
Modulus squared

analytic continuation to the complex
plane
Correlation
P1 and P2 hold
Fourier Transform
Inverse Fourier Transform
4
Analytic Continuation and Fourier Tools II.
Fourier Transform

P1 reads
Conformal change of variable
Inverse Fourier Transform
P2 guaranteed by the conformal mapping
P3 see next slide
Fourier Transform
Inverse Fourier Transform
5
Invariance Property of the Spectrum

P3 C(k) C(- k) implies the symmetry of the
two parts of this formula
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Natural Spectra in the physical sense
Violin Modes Mathematical Tools for Noise
Analysis E. Cuoco A. Vicere 1994

meromorphic function Infinite number of simple
poles in the variable f, located on a cone in the
complex f domain one simple pole at the
origin.
Filtered thermal noise A thermal noise linearly
filtered by the machine will produce rational
spectra
Opto. Thermal couplings in mirrors Vinet 2001
It is reasonable to look for meromorphic or
rational approximants of spectra even in the
presence of branch points, in view of the good
behavior of Padé Approximants.This is what ARMA
models do we will do it the other way around
7
Natural form of the Spectrum in the
mathematical sense

Finite number of poles or zeros in a finite domain
This must be true for the unit disk D1, which
contains A poles and B zeros
Through (I), the same is true for the domain
outside the unit disk
(Th)
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Symmetrized Padé Approximant I.

9
Symmetrized Padé Approximant II.

Lesson from the numerical simulations to come
as compared eg to AR, the robustness and
practical value of the method dwell on the more
careful representation of the poles -the zeros
of Q and Q - which has been obtained using the
classical Padé Approximants