# 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.

Pade

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.

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1994

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