arXiv subjects
Publications and source records attributed to Jean-Daniel Fournier.
This report aims to provide gravitational waves data analysts with an introduction to the ideas and practice of the Pad\'e Filtering method for disentangling a signal from the noise. Technically it comes to the tracking of the zeros and singularities of random z-Transforms by noisy Pad\'e Approximants.
In the context of star capture by a black hole, a new noticeable difference between Brans-Dicke theory and general relativity gravitational radiation is pointed out. This feature stems from the non-stationarity of the black hole state, barring Hawking's theorem.
The average density of zeros for monic generalized polynomials, $P_n(z)=ϕ(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $ϕ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $δ= \frac 1{\sqrt{nλ}}$. The average density tends to a simple, {\em universal} function of $ξ={2n}{\log |z|}$ and $λ$ in the domain $ξ\coth \fracξ{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.