Padé and Symmetrized Padé Approximants in Data Processing

1. Padé and Symmetrized Padé Approximants in Data Processing Jean-Daniel Fournier (1)(2) member of the CNRS fournier@obs-nice.fr Bénédicte Dujardin (1) (2)(3)(4) Ph.D. student dujardin@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“

2. 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

3. 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 P2 guaranteed by the conformal mapping Inverse Fourier Transform 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

6. “ Natural“ Spectrain 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 Spectrumin 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)

8. 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