3.2 Lemniscates, Capacity, and Measure

The purpose of this section is not to give a detailed and rigorous account of the foundations of capacity and measure, but rather to indicate why the results of the previous section are much stronger if rephrased in terms of capacity or Hausdorff measure.

 

and M is sufficiently large,

 

Hence, provided

 

The basis of Section 6.5 is the result related to Cartan's lemma [Cartan. 1928; Nuttall, 1970b] that, for a polynomial qm{z) with leading coefficient unity, | qm(z)\ >t]m except on a set & of measure at most 7ttj2. The boundary on which | qm(z)| =fjm is usually called a lemniscate, and so all the results of

Section 6.5 apply except on lemniscatic regions which are arbitrarily small. We will see that the natural measure of the size of a lemniscatic region is its capacity.

To begin with, let us recall Tchebycheff's result for the minimax poly­nomial on an interval The problem is to find the polynomial Tn(x) in the class Pn of all polynomials p„(x) of degree n and leading coefficient unity for which the limit

 

 

is attained. The solution is well known. For «>1,

 

 

is a polynomial of degree n of leading coefficient unity, and

 

 

A linear change of variable

 

 

leads to the result that

 

 

To generalize these ideas to an arbitrary compact set in the z-plane, we have the following theorem.

Theorem 6.6.1. Let & be a compact set in the complex plane (containing infinitely many points). Then there is a unique Tchebycheff polynomial for which

 

 

may be written as

is attained. The minimax polynomial,

 

 

 

 

 

The zeros

 

is achieved at least n times on the boundary of &.

lie in the convex hull of $, and the maximum value Mn of

Discussion. We shall not prove this important theorem, [Hille, 1962, p. 265], but elaborate it with a few remarks.

is attained at z = z', we find that

 

 

It is easy to see that all the z, lie within the convex hull of &. For suppose not, and let z[ lie outside the convex hull DC of S and the points z2, z3,..., z„. By considering a point z| nearer to %, and the point z' for which the

 

 

which contradicts the minimum property of Mn with respect to variation of the zr Hence the zeros of Tn(z) lie in the convex hull of S.

The proof of the existence of a minimax polynomial is based on "tracing" the roots to extremal positions in the closed convex hull. We omit the proofs of existence and uniqueness of Tn(z) in the general case. Since & is compact, it is obvious that the maximum of \Tn(z)\ is attained in S. Further, the maximum-modulus theorem shows that the maximum is attained on the boundary of &. That \Tn{z)\ should equal Mn at n distinct points on the boundary is another significant result we state without proof.

If $ is a finite point set, a possibility excluded by the hypothesis of the theorem, then the Tchebycheff polynomial of order n is zero on & for n>N. Furthermore, it is not unique for n>N, and so this degenerate case is naturally excluded by the hypothesis.

 

Figure 1. A set S = S, US2. The union of the shaded region and the set £ comprises the convex hull of &.

Corollary. The Tchebycheff polynomial Tn{z) for a set & defines a lemniscate by \Tn{z)\ = Mn and a lemniscatic region £n by

 

 

Then Sc£„, and the boundary of £„ has at least n points in common with S.

Discussion. The important idea is that & becomes a subset of the lemniscatic region £n defined by \Tn(z)\<Mn, and the proof follows im­mediately from the maximum-modulus theorem and the main theorem.

Theorem 6.6.2. Let & be a compact set in the complex z-plane (containing infinitely many points). Let Tn(z) be the Tchebycheff polynomials defined on $, and let

 

Then the capacity of & is uniquely defined by

 

 

 

 

 

Hence

We now proceed to three theorems which we prove, because the proofs illustrate the structure of lemniscates and the idea of capacity. Repeated use is made of the maximum-modulus theorem, which states that the maximum modulus of an analytic function in a compact region is achieved on the boundary.

Proof. The only proof required is that the limit (6.7) is well defined. To do this, let

If 8 is the maximum diameter of the compact set S in the usual sense, then from (6.4),

Given e>0, we may find N such that

 

 

 

 

 

and

 

 

 

 

 

For any positive integers m, k but with m<N,

 

 

 

 

 

where

 

 

 

 

is a constant independent of k. Recognising zm[Tn{z)]k as a polynomial of order m + nk, we see that

 

 

and by taking the limit as k^> oo,

 

 

Equation (6.8) holds for any positive m<N, and any e>0. Hence /8 = a, and cap(S) is well defined by (6.7).

The quantity cap(S) is a measure of the magnitude of the set &. It is called the logarithmic capacity, colloquially abbreviated to capacity or transfinite diameter in different contexts.

 

is given by

defined on 8, so that

le the «th-order Tchebycheff polynomial

 

The next theorem shows the key role of lemniscates in the theory of capacity.

Theorem 6.6.3. If & is a lemniscatic region given by

 

 

 

 

then

 

Proof. The maximum modulus theorem show that 3S (the boundary of S)

 

then

 

 

and consequently

 

By definition of

 

 

Recall

RoucHfi's theorem. If /(z) and g(z) are analytic inside and on a closed contour C and |g(z)|<|/(z)| on C, then /(z) and /(z) + g(z) have the same number of zeros inside C.

In this case f(z) = Tn(z), g(z)= —/>„(z), and the theorem implies that a polynomial of degree n— 1 has n zeros in $. This is impossible, hence tj'=tj and cap(S)=rj, proving the theorem.

Before giving examples of the capacity of a set, we prove the major result which improves the theorems of the previous section.

 

where

We will consider the family of lemniscates

given by \p(z)\ = p for all positive p. This consists of at most n disjoint closed curves K>, K2,..., K . The maximum-modulus theorem shows that

Theorem 6.6.4. Let & be a compact set. Then

 

 

Proof. Let Tn( z) be the «th Chebychev polynomial denned on &. ror any e>0, (6.7) implies that a sufficiently large n exists such that

 

 

This inequality defines a lemniscatic region £„, of capacity -q = cap( &) + e, which has & as a subset. If we prove that

 

 

it follows that

 

 

Hence the theorem is true provided we prove (6.10). For brevity, let the lemniscate be defined by

 

 

Figure 2. The mapping of K, and K2 onto the circle |f| = p, showing two branch points in the f-plane.

 

the curves lie outside each other. There is at least one root of p{z) in each

curve K. • 1ft thfirp hp nrprisplv ii . rnnts in K. As thp r>r>int 7 mnvps arnnnrl

 

moves once around the circlt

 

and

moves a,, times around the circle

 

mapping

 

 

and the inverse

Herep(z) is clearly a single-valued function of z, but

Now consider the mappings

 

 

 

the inverse map z=z(f) is not, and z=z(J) has n— 1 branch points occurring at p'(z) = 0. However, assuming that |?| = p avoids these branch points, one of the branches of z=/(f) is regular on |f'| = pl/llk, and has the Laurent expansion

 

i.e.,

 

 

 

(6.12)

 

 

To find the area of Kk,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence

Therefore

 

. (6.13)

 

Equation (6.13) is an increasing and continuous function of p, and so is bounded by its value at p= oc. For p sufficiently large, Kl contains all n roots of d(z\ and therefore a,=n and m = 1.

 

 

 

 

 

(6.14)

which makes (6.12) more explicit, and (6.13) becomes

The result is proved, but it is interesting to note from (6.15) that equality is attained when a0=fli = ■ ■ ■ =0 and (6.14) shows that this locus is the circle z" = p.

We now turn to a few examples.

Example 1. The capacity of the interval a<x<b is (b—a)/4.

Discussion. We assume the result (6.1) [Cheney, 1966, p. 61; Rivlin, 1969, Chapter 1] for the proof. Then result (6.3) follows from the substitu­tion (6.2), and from (6.16) it follows that

 

 

and the theorem is proved.

Example 2. The capacity of the disk | z | < R and the circle | z | = R is each equal to R.

Proof. The circle |z| = i? is a lemniscate, given by \zm\ = Rm, and so the results follow from Theorem 6.6.3 (and also from an inspection of the proof of Theorem 6.6.4).

Example 3. If & is a countable compact set, cap(S) = 0. We leave the proof as an exercise.

We next state two very important theorems about capacity, without proof. Each gives considerable insight into the magnitude of the capacity of an arbitrary point set.

 

Discussion. The points

ire chosen so as to maximize

 

Theorem 6.6.5 [Hille, 1962, p. 268-273]. Let & be a compact set. Then

 

 

which contains n(n— l)/2 terms in its expansion. Hence it is clear that cap(S) is bounded by the maximum diameter of 6. Furthermore, cap(S) is some sort of geometric mean of the distance apart of the points of $, and so it is called the transfinite diameter.

Two corollaries of the theorem are self-evident:

Corollary 1 (Monotonicity). If <3) and & are compact sets with йі!С&,

 

 

Corollary 2 (Homogeneity).

 

 

 

 

Theorem,6.6.6 [Hille, 1962, p. 280-289]. Let ju(z) be a normalized measure defined on &, and define

 

 

Let

 

 

Then

 

 

Discussion. We have in mind that ju(z) is a charge distribution on a two-dimensional surface S, so that I[ji] is the self-energy associated with the distribution ji. In physical equilibrium, this functional is a minimum, and so the potential due to the physical distribution of unit charge on $ is

 

 

The further definition that ln(cap(S)) = - V{&) explains the name of loga­rithmic capacity for cap(S).

Finally, we remark that we have emphasized the role of capacity as a point-set measure of the region of inaccurate approximation of Pade ap­proximants to meromorphic functions. Another popular point-set measure is Hausdorff measure, and the results of the previous section generalize to convergence in a-dimensional Hausdorff measure [Wallin, 1974; Lubinsky, 1980]. In this case, the measure is defined by

 

 

where the minimum is taken over all possible denumerable families of circular disks Dt which cover the set S, and S( Dt ) is the diameter of the disk Dr Two-dimensional Hausdorff measure is similar to area in its properties, and one-dimensional Hausdorff measure is more like length.

The following Boutroux-Cartan lemma is most useful in the context of one-dimensional Hausdorff measure.

 

satisfies

 

Theorem 6.6.7. For any v>0, the polynomial

the inequality

We do not prove this theorem, and we refer to EPA (Chapter 14) for a more complete account of the role of Hausdorff measure in convergence of Pade approximants. We merely note that (6.16) is the basic type of inequal­ity required in this context. We also note that many of these result, also extend, mutatis mutandis, to the rational interpolation problem [Walsh, 1969, 1970; Karlsson, 1976].

We conclude this section by repeating that an inspection of the proofs of the theorems of Section 6.5 shows that they prove convergence in capacity, and we have shown that this is a substantially stronger result than conver­gence in measure. In particular, an interval has zero measure, but nonzero capacity (and nonzero one-dimensional Hausdorff measure), and so a finite interval, or any set containing a finite interval, is not a permissible excep­tional set for the theorems of Section 6.5. Finally, we draw attention to a result of Nuttall's, possibly generalizable in the future, that for certain functions with branch points connected by branch cuts, the Pade approxi­mants converge in capacity in the z-plane except in a region which mini­mizes the capacity of the cuts in the z ~ 1 -plane [Nuttall, 1977; Nuttall and Singh, 1977]. This conjecture should be contrasted with the conjecture of Baker, Gammel, and Wills.