Maciej Pindor

Padé approximants and noise: rational functions

Abstract We continue a study of Pade approximants (PA) for a series perturbed by random noise – this time we consider more general rational functions. We begin with the simple case of a sum of two geometric series, and then show how these considerations can be extended to a general rational function. We do not study the most general case, but rather concentrate on demonstrating how our results for geometric series extend to new situations encountered when a general rational function is considered. We show that Froissart doublets are a universal feature and we construct an analog of the Froissart polynomial introduced in the earlier paper.

Continued fractions, two-point Padé approximants and errors in the Stieltjes case

A Stieltjes function is expanded in mixed T- and S-continued fraction. The relations between approximants of this continued fraction and two-point Pade approximants are established. The method used by Gilewicz and Magnus (J. Comput. Appl. Math. 49 (1993) 79; Integral Transforms Special Functions 1 (1993) 9) has been adapted to obtain the exact relations between the errors of the contiguous two-point Pade approximants in the whole cut complex plane.

An analysis of mean life and lifetime of unstable elementary particles

A theoretical analysis of the concept of lifetime and mean life of unstable elementary particles is presented. New analytic formulas for lifetime and mean life as a function of decay width Γ and the mass of unstable particle are derived for Breit-Wigner and Matthews-Salam energy distributions. It is demonstrated that, for unstable particles with a larger width or decay energy threshold, the deviation from the generally accepted mean life τm =Γ−1is significant. The behavior of the decay law P(t) for small times is analyzed, and it is shown that the Breit-Wigner distribution violates the condition P(t = 0) = 0, whereas the Matthews-Salam distribution satisfies it.

THE FOOTPRINTS OF NOISE IN RATIONAL INTERPOLATION

The reliability of rational approximations built from noisy data is explored. One considers a rational interpolation f(t) of a function S(t) known only via its, randomly perturbed, values at a finite number of real points. It is found that in the small noise amplitude regime, the zeros and poles of f(t) split into two families: one reflecting the analytic structure of the function S(t); and one made of coalescent zero-pole pairs, most of the time confined in the real and/or complex vicinity of the interpolation interval. This exciting result seems to open the door to the detection of noise and, perhaps, to its subsequent erasure.

Rational Interpolation from Stochastic Data: A New Froissart's Phenomenon

AbstractThe 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

Rational Approximation and its Applications in Mathematics and Physics

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Pade Approximants and Borel Summation for QCD Perturbation Expansions

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