5.3 Moment Problems and Orthogonal Polynomials

 

The principal question left open in the previous section is what the [M+J/M] Padé approximants of Stieltjes series converge to. This question is part of a wider question: to what extent do the coefficients fj determine the measure d<t>(u)l The latter is a moment problem. If it is true that for some positive measure d<t>(u) defined on — oo<«<oo the coefficients

(3.1)are finite and well defined, then it is natural to call f, the moments associated with the measure. The phraseology has a historical setting in which <j>'(u) is a density per unit length, necessarily positive, of a linear mass distribution, such as a beam. If the beam has variable density, 4>'(u) is not constant. If the beam has a weight attached at <j>(u) has a positive jump discontinuity at The integration limits in (3.1) are set by the length of the beam, outside of which d<j>(u) = 0. The mathematical questions which emerge from this physical setting have the name of moment problems. Given the values of all the moments fj,j=0,1,2,..., the problems are

Existence. Does a positive measure d<j>(u) exist to allow the repre­sentation of {fj} by (3.1)?

Determinacy. Is d<t>(u) uniquely determined?

Nature. Are the {fj} Stieltjes moments? Are they Hamburger mo­ments?

The answers to these problems depend on various conditions on the moments. We will explain the solutions as well as the problems in their various settings. Stieltjes gave a clear answer to some of the outstanding problems in the form of the following theorem.

 

satisfy the determinan-

 

 

Theorem 5.3.1. If the coefficients

tal conditions

then a Stieltjes

 

measure d<j>(u) exists for which

 

 

Further,

 

 

is a Stieltjes series with the given coefficients {fj} in its formal expansion.

Proof. An inductive proof based on Sylvester's identity establishes that D(m,n)>0 for all m,n>0 given that D(0,n)>0 and D( 1, n)>0 for all n.

Consider the sequence of [M—l/M] Pade approximants to the formal power series

 

 

Theorems 5.2.1, 2, 3, 4, 6, and 7 all concern Pade approximants and are based entirely on properties of the coefficients fj — in fact, the properties

that the determinants D(m,n) are positive. Consequently, Theorem 5.2.6 i* based on valid hypotheses, and the [M— 1 /M] Pade approximants of (3.4 > converge uniformly to an analytic function /(-l)(z) in ^(A) shown in Figure 3 of Section 5.2. We deduce from Theorem 5.2.1 that

 

for

 

 

 

with

 

(3.5.

 

 

 

 

For given A>0, we may let M—> oo, giving

From (3.5),

Let us apply Cauchy's theorem to [M— 1 /M] using a contour C which i> the boundary of ^(A), and inside which [M— l/M](z) is analytic. This contour is shown in Figure 1, and Cauchy's theorem states that

 

 

We may now take the limit as > oo, noting that the contribution from the large circle tends to zero, and obtain, using (3.8) and (3.9),

 

 

 

 

 

For any open interval (u, v) of the negative real axis, we may choose

 

 

 

 

 

and consider

 

 

 

 

provided the limit is well defined. The details are given in Section 5.6. Equation (3.12) provides a construction of a Stieltjes measure so that (3.11) may be written as

 

 

for z not on the negative real axis, — oo<z<0. We have avoided any discussion of the value of <f>(u) at a jump discontinuity, because this value isusually of no importance, and we say that <j>(u) is substantially determined by (3.12). Equally, we may take <p(0) = 0 without loss of generality. The theorem is now proved.

It is important to realize that this theorem answers the questions of existence and nature, but not the question of uniqueness. Proofs of unique­ness are presently based on further hypotheses, as we discuss in Sections 5.4 and 5.5.

Let us now consider a change of variable w=— z"1 which reveals a remarkable connection between Pade approximants and orthogonal poly­nomials. It is, in fact, the historical approach to orthogonal polynomials, and rightly so.

 

for

We consider the basic initial representation

 

 

and rearrange it and define F(w) by

 

 

which has the formal expansion about w= oo,

 

 

If we construct a set of polynomials {wm(«), m = 0,1,2 • • •} orthogonal over dé>(u), we fundamentally require that

 

 

This equation is tantamount to the orthogonality condition

 

 

and we have taken the usual assumption that each irk(u) is a polynomial of degree precisely equal to k for granted. By writing

 

the equation (3.17) becomes

 

 

 

and by taking

 

this set of linear

With the identification

 

equations is seen to be the Padé equations (1.1.6), with the solution

Hence (3.19) becomes

 

 

 

Since

 

but conventionally normalized orthogonal polynomials given by

and

for Stieltjes series, we prefer to use the equivalent

 

 

These polynomials irm(u), defined by (3.20), satisfy the orthogonality condi­tion (3.18). A natural observation at this point is that the (***) identity (Section 3.5) for three consecutive Qlm~ l/m,(z) denominators immediately is interpreted as the recurrence relation for the orthogonal polynomials. As a corollary to this development, we observe that the set are orthogonal polynomials over

Next, we proceed with the converse development. We assume the usual properties of the orthogonal polynomials {irm{u), « = 0,1,2,...} and define polynomials (3.30) whose ratio is the Padé approximant of /(z) defined by (3.14).

 

To determine the numerator of the Padé approximant to/(z), or of F(w) expanded about w=oo, recall (3.15),

 

 

 

which leads us to expect

 

 

 

where pm(w) is a polynomial of degree m— 1. Thus we are led to consider

 

 

 

 

(3.23)

 

Equation (3.23) splits into two parts, and we find

 

 

 

 

 

(3.25)

 

 

A glimpse forward to (3.30) explains why it is convenient to use a subscript m for a polynomial of degree m — 1 in this instance. The second part of (3.24) is

 

 

 

 

 

 

 

 

 

 

(3.26)

 where the orthogonality property (3.17) has been used. From (3.23),

 

(3.27)

Using the O-notation in the sense of formal series operations, we find from(3.20) and (3.26) that

 

 

 

 and hence from (3.26)

 

 

 

Recalling from (3.25) and (3.20) that pm(w) and irm(w) are polynomials of degrees m — 1 and m respectively, we have proved that

(3.29)

where

 

Hence, following (3.20), we define

 

 

 

(3.30)

 Hence we see that, except for normalization, p^m~x/m\z) and Qlm~l/m](z) are the numerator and denominator polynomials of Pade approximants. Furthermore, for (3.15), (3.24), (3.26), (3.30), and (3.31) we have the explicit error formula

 

 

Other formulas of this kind are given in Part II, Section 3.1.

with pm, 77m defined by (3.19) and (3.25). This proves that

 

Equations (3.14)—(3.31) show a different approach to the construction of Pade approximants. As a bonus, (3.25) indicates that the polynomials pm(w) satisfy exactly the same recurrence relation as wm(>v), the proof needing no more than the orthogonality property (3.17).

For general reviews of the scope of this section, we refer to Allen et al. [1975], and Karlson and von Sydow [1976], who show that much of theor> of Sections 5.1-5 can be derived using orthogonality methods.

Exercise 1. The Laguerre polynomials Ln(u) satisfy an orthogonality condi­tion of the type (3.18). Deduce that the associated Stieltjes function is given bv

 

 

where £,(z) is defined by (4.6.8) in |arg(z)|<w. Use (3.21) and (3.25) to find the [m~ 1 /m] Pade approximant of /(z).

 

Exercise 2. Only the first 2M+1 moments, /„. /,..... f2M of a Stieltje> density (1.2) are given. Use the formal expansions

 

 

to prove that the unknown moments are bounded by

 

for