# 5.4 Stieltjes Series Convergent in |z|<i?

If the Stieltjes series discussed in the previous sections have a nonzero radius of convergence, which is to say that they are analytic in a neighbor­hood of the origin, then convergence theorems are easily proved and the moment problem is determinate. This section is devoted to the results ensuing from the hypothesis that /(z) is a Stieltjes series with a nonzero radius of convergence, and they may be contrasted with the results of Section 5.5.

The property that the poles of the Pade approximants of f{x) are on the cut of /(z) is retained. In fact, we have the stronger result:

Theorem 5.4.1. Let f{z) be a Stieltjes series convergent in jzj<R. Then the poles of the [M+J/M] Pade approximant, with 7> — 1, lie on the real axis in the interval — oo <z< — R.

Method 1. Suppose the contrary. From Theorem 5.2.1, there is a pole at z = z0 with — /?sSz0<0 of the [A/, +7,/A/,] Pade approximant, with 7, — 1. The interlacing property implies that every [M+J^M] Pade ap­proximant with M>M\ has a pole in the interval (z0,0), and let the limit of

the poles nearest to z = 0 be at z = z,. Then from Theorem 5.2.7,

Hence, |z,|>/?, contradicting the hypothesis. Therefore, the poles of the Pade approximant lie on the open interval (— oo, — R).

for

Method 2. If /(z) is analytic in |z|<i?, its Stieltjes-integral representa­tion can have no singularities in this circle, and so becomes

We may also take

The poles of the [m+J/m](z) Pade approximant occur at zeros of wm(w), where z= — u~x and ^(w) is a polynomial satisfying the orthogonality conditions (3.17):

Suppose that m, zeros of mm(u) do not lie in 0<u<R but elsewhere in the complex w-plane at u = u],u2,..., wm|. Then wm(w) has the representation

where k is a normalization constant. Consider

The integral of / is strictly positive, having no sign changes at the points wm +1,....r»m, where it vanishes. But, by orthogonality, 7=0. Thus m,= 0, all zeros, of -nm(u) lie in (0, i?-1), and all poles of the [M+J/M] Pade approximant lie on the cut of /(z).

The next theorem concerns the limit functions f<J\z) of the paradiagonal sequence [M+J/M] of Pade approximants of a Stieltjes series /(z), which are shown to be identical to /(z). Again, we give two methods of proof, based on the integral representation and on orthogonal polynomials.

Theorem 5.4.2.

approximants with

is analytic in

be the limit functions of

to a Stieltjes series f(z), which are analytic in

Texts on Computational Mechanics

Volume VII

Series Editor: JOHN ARGYRIS, F.R.S.

EDITORIAL BOARD

Kari APPA, Northrop Company, Hawthorne CA, USA Herman BERENDSEN, University of Groningen, The Netherlands David BURRIDGE, ECMWF, Reading, UK T. J. CHUNG, University of Alabama, Huntsville, AL, USA Ioannis St. DOLTSINIS, ICA, University of Stuttgart, Germany Jean DONEA, ISPRA, Varese, Italy Egon KRAUSE, Aero Institute, Aachen, Germany Roger OHAYON, ONERA, Paris, France

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An Exploration of Chaos

JOHN ARGYRIS GUNTER FAUST MARIA HAASE

An Introduction for Natural Scientists and Engineers

1994

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