Jacek Gilewicz

Padé approximants and noise: a case of geometric series

Abstract We study effects, on Pade approximants (PA), of a random noise added to coefficients of a power series. Numerical experiments performed during the last 30 years have shown that such noise results in the appearance of, the so-called, Froissart doublets (FD) — pairs of zeros and poles separated by a distance of a scale of the noise. For the first time we can prove that this effect actually takes place for geometric series. We can also show what happens with noise-induced poles or zeros that do not form FD. We construct a polynomial that has as its roots mean positions of FD. Coefficients of this polynomial are expressed by random perturbations of coefficients of the series. We study numerically this polynomial for low orders to show where (in these orders) FD coalesce. Numerical study of this polynomial, to see a behaviour of FD for large orders, is much easier than a study of the entire PA. We also express a deviation of a pole of PAs approximating the true pole of 1 (1 − z) , from 1, by perturbations of coefficients and show numerically that when order of PAs increases, this deviation drops down.

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.

Froissart doublets in Padé approximation in the case of polynomial noise

First, we study the relation between the zeros of random polynomials Rn+1 and the zeros and poles of their Pade approximants [n/n]Rn+1. Next, we consider the distribution of zeros and poles of Pade approximants to the geometric series perturbed by a random polynomial noise. We observe numerically interesting connections between two above problems. Some numerical observations on the Froissart doublets have been also made.

Optimal inequalities of padé approximant errors in the stieltjes case: closing result

For a nonrational Stieltjes function we prove the inequalities involving all ita Pade approcimants (i.e.m and n ≥ 0). Slightly more accurate inequalities containing the Pade denominators values are also given. This completes and sharpens known inequalities established by Wynn, Baker, Brezinski, and also settles one of our conjectures

Widths and shape-preserving widths of Sobolev-type classes of s -monotone functions

Let Ibe a finite interval, r, n ∈ N, s ∈ N0 and 1 ≤ p ≤ ∞ Given a set M, of functions defined on I denote Dy Δ+ s M the subset of all functions y ∈ M such that the s-difference Δτs y (ċ) is nonnegative on I, ∀ τ > 0. Further, denote by Wpr the Sobolev class of functions x on I with the seminorm ||x(r)||Lp ≤ 1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes Δ+s Wpr in Lq for s > r + 1 and (r, p, q) ≠ (1, 1, ∞). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically = n-2.

Location of the zeros of polynomials satisfying three-term recurrence relations. I. General case with complex coefficients

The finite sequences of polynomials {Pn}n = 0N generated from three-term recurrence relations with complex coefficients are considered. First a general method is presented which allows the determination the regions where all zeros of the polynomials in question are located. Next one way is followed, say ¦μn¦ ¦βn¦, is followed. Subsequent papers (E. Leopold, J. Approx. Theory43, 15–24 (1985); E. Leopold, Location of zeros of polynomials satisfying three-term recurrence relations. IV. Application to some polynomials and to generalized Bessel polynomials, in preparation) are devoted to some particular cases and to numerical applications.

Some cubic birth and death processes and their related orthogonal polynomials

AbstractThe orthogonal polynomials with recurrence relation (λ,n +μn-z)Fn(z) = μn+1Fn+1(z)+λn-1Fn-1(z) with two kinds of cubic transition rates λn and μn, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.

Approximation of the Integrals of the Gaussian Distribution of Asperity Heights in the Greenwood-Tripp Contact Model of Two Rough Surfaces Revisited

Some examples of application of Pade approximant techniques to approximate the integrals in question and comparison with previous results, essentially the recent Panayi-Schock and Green-wood results, show the efficiency of this simple natural approximation.

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.

Location of the zeros of polynomials satisfying three-term recurrence relations with complex coefficients

Two papers on the same subject were pubkushed by the authors in 1985:part I[1]and part III[2].The method announced for part II did not give satisfying results.However,we noticed that our method of estimationg the boundaries of zero-free regions for polynomials can be improved by introducing proper optimizing paramcters for each polynomial.The optimization is then performed over all these parameters at the end. It was showed that this finest method of optimimation is sharp in the following sense:it leads to the exact interval where all zeros of classical orthogonal polynomials are licated[3].In the present paper we give the complete proofs of these improved theorems of lication in quaestion.