Rational Interpolation from Stochastic Data: A New Froissart's Phenomenon
The analytic structure of Rational Interpolants (R.I.) f(z) built from randomly perturbed data is explored; the interpolation nodes xj, j = 1,...,M, are real points where the function f reaches these prescribed data \(\widetilde\phi _j\). It is assumed that the data \(\widetilde\phi _j\) are randomly perturbed values of a rational function ϕ(n)(m) (m / n is the degree of the numerator/denominator). Much attention is paid to the R.I. familyf(n+1)(m−1), in the small stochasticity régime. The main result is that the additional zero and pole are located nearby the root of the same random polynomial, called the Froissart Polynomial (F.P.). With gaussian hypothesis on the noise, the random real root of F.P. is distributed according to a Cauchy-Lorentz law, with parameters such that the integrated probability over the interpolation interval ⌈x1, xM⌉ is always larger than 1/2; in two cases studied in detail, it reaches 2/3 in one case and almost 3/4 in the other. For the families f(n+k)(m+k), numerical explorations point to similar phenomena; inspection shows that, in the mean, the localization occurs in the complex and/or real vicinity of the interpolation interval.
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