5.1 Introduction to Stieltjes Series

 

Definition. A Stieltjes function is defined by the Stieltjes-integral" representation where <p(u) is a bounded, nondecreasing function (taking infinitely man\ different values) on 0sg«<oo and with finite real-valued moments given b>

 

 

From (1.1), it follows immediately that /(2) is a real symmetric function, defined in the cut z-plane with the cut along the negative real axis as shown in Figure 1; real symmetric functions are defined by (1.6).

A formal expansion of (1) always provides a series expansion of f(z 1. called a Stieltjes series, and given by

 

 

1 he series is called formal because it may not converge for any z (except z = 0); nevertheless it is a useful representation of the function /(z). if

*A good explanation of Stieltjes integrals is given in Perron [1957, Vol. 2, p. 180] or Rudin [1976, Chapter 6].

 

Figure 1. The cut z-plane in which f(z) is defined by (1.1).

 

 

standard notation,

because the determinantal inequalities of

properly reinterpreted, as we will show in this chapter. It is easier to use the positive definite coefficients {fj} in the expansion (1.3) rather than our

Theorem 5.1.2 take on a simpler form.

 

for

 

 

 

with

Hence /(z) is a rational function of z with m simple poles at z= — w"1 on the negative real axis and with positive definite residues. Furthermore, all Pade approximants of /(z) with L^m— 1 and M^m are exact. Thus, the case when <j>(u) takes a finite number of values only is a special case, and it is usually excluded, by definition, from being a Stieltjes series.

The phase "taking infinitely many different values" in the definition of a Stieltjes series is made part of the definition so as to exclude the following special case. If §(u) takes on a finite number of values, say m+ 1 distinct values, then <j>(u) is piecewise constant onm+1 intervals covering the range OsSw<oo. Suppose that

 

 

 

 

 

 

Then d<j>(u) = 0 except in neighborhoods of w = w,, i= 1,2,..., m, and so

 

 

 

 

In this case, /(z) is defined in the cut z-plane, cut along — oo<z< — A The power-series expansion of /(z) is given by (1.3) is then convergent in the disk

 

 

convergence, the Pade approximants of the series are vital for its analysis and are useful for its numerical evaluation, as we will see in this chapter. We can prove convergence of the Pade approximants largely because we can prove that the poles of the Pade approximants lie on the cuts of the Stieltjes function. Stieltjes functions are real symmetric functions defined in the cut plane, with the negative real axis as the cut.

shown in Figure 2.

Whether or not the formal series

 

has a zero radius of

If §(u) is constant on \s£w<oo, then

A function is defined to be real symmetric if it takes complex conjugate values when the variable is complex-conjugated [Titchmarsh, 1939, p. 155].

 

Figure 2. The cut z-plane in which/(z) is defined by (1.5).

 

 

This condition is that

(1.6)

An important and immediate consequence of this applies to a function f(z) which is analytic at a point z = x0 on the real axis, so that the expansion

 

 

is convergent in some small disk enclosing z = x0. The coefficients dn / = 0,1,2,..., are real if and only if f(z) is real symmetric, which justifies the name "real symmetric".

Stieltjes functions are real symmetric and, as can be shown from (1.1), have a negative imaginary part on the cut in the sense that

 

 

(1.7)

provided the implied limit (e—>0) exists (cf. Lemma 3 in Section 5.6). We distinguish the three cases (i) tp(u) is differentiable, (ii) tp(u) is continuous but not differentiable, and (iii) (/>(w) is discontinuous at a point u [Riesz and Nagy, 1955].

Before embarking on the proofs of the properties of Stieltjes series, let us consider an illustrative example a Stieltjes series. The function is

 

 

 

 

 

Its coefficients are given by

 

 

Hence the density function defined by

 

 

 

 

(1.8)

 ensures that

 

(1.9)Further,

 

 

so that f(z) is a Stieltjes series according to (1.1), (1-2), where <j>(u) is a nondecreasing function taking on infinitely many values [given by (1.8)] and all the moments are well defined by (1.9). The real symmetry is evident from the real coefficients in the expansion, and also

 

 

 

 

illustrating (1.7).

The following theorem shows the effect on a Stieltjes series of deletion of the first J terms of the series.

 

 

 

Theorem 5.1.1. Let f(z) be a Stieltjes series with a formal expansion ana representation

 

 

Then g(z) given by the formal expansion

(1.10)

is also a Stieltjes series represented by

(1.11)

Proof. The results follow immediately from the definitions (1.1)-(1.3).

 

Stieltjes series may be recognized by virtue of the determinantal condi­tions satisfied by the power-series coefficients fj. We use the following definition, which is more convenient than using the determinants C{ L/M ) defined in (1.4.8) in this context:

(1.12)

These definitions are related by the identities (see exercise 1)

 

 

 

 

 

 

 

 

 

Theorem 5.1.2. A necessary condition for f(z) to be a Stieltjes series is that all the determinants D(m, n) with m>0, n >0, are positive.

Remark. The condition is also sufficient, provided that f(z) is uniquely determined by its power series in principle. The proof of this result follows later.

 

 

Proof. We use the properties of Stieltjes series

 

 

and we must prove that the coefficients fj satisfy D(m, n)>0. Notice that each coefficient fm is defined to be positive, and so D(m,0)>0. Let us define

 

 

 

 

 

 

 

 

 

which is seen to be a real positive quadratic form in the n + 1 real variables Xq j X |, . . . ^ X „ • Therefore G(x0,...,xn) has a minimum value on the hyper- sphere

 

 

The values of the {xj at this minimum are given by Lagrange's unde­termined-multiplier method, following variation of the n independent coor­dinates. The equations are

 

 

and S(x)= 1.

 

(1.161

which is a set of n +1 homogeneous equations for x()... xn with a con­sistency condition that A must be an eigenvalue of the real and symmetnc matrix

(1.171

Let x(k) be the eigenvector associated with the eigenvalue X(k). Then we find from (1.13) that

 

 

and from (1.14 ) that