3.3 The Pade Conjecture

A conjecture of Baker, Gammel, and Wills, slightly rephrased, is popu­larly known as the Pade conjecture. The conjecture, which concerns conver­gence of diagonal Pade approximants to functions analytic in a disk, gave great impetus to the search for convergence theorems for the diagonal sequence, and also led to confidence in the usefulness of the diagonal sequence.

 

which is valid in the z-plane outside no more than n circles of radii rt which obey the inequality

 

Before commencing the statement of the conjecture, we present Gammel's counterexample [Baker, 1973b] which shows why the conjecture takes its form and is not stronger.

Gammel's counterexample. Let

 

 

 

revealing a block of length nk. This result implies that

 

where the indices nk are defined by = 1, nk+l =2nk + 1. Let cn=akzk " if n is such that nk<n<nk+v The coefficients {a^} are specified by

 

 

This choice of ak ensures that

 

 

so that /(z) is holomorphic (by the comparison test). The function may also be expressed by

 

 

Inspection of (7.1) and the [L/1] sequence calculated by the accuracy through order method shows that

(as is obvious by accuracy-through-order criterion anyway), and so [nk/nk] has a pole at z=zk. By selecting the sequence {zk, k= 1,2,...} to be a set of points dense in the plane, and allowing such repetition as is necessary, we construct an analytic function with a subsequence of diagonal Pade ap­proximants (in fact [2* —1/2* —1] Pade approximants) which diverges everywhere in the plane. In fact, a host of Gammel counterexamples can be

constructed with Pade approximants which diverge in any desired region of the z-plane.

The only redeeming features of Gammel's counterexample are that the area of bad approximation at the poles of the [nk/nk] Pad6 approximants tends to zero rapidly with increasing and that Baker has shown that another subsequence of diagonal approximants converges pointwise [Baker, 1973b],

Consideration of the implications of the counterexample shows why the Pade conjecture which follows is widely believed to be both true and as strong a result as possible.

Conjecture [Baker et al., 1961]. Let f(z) be analytic in \z\<R except for m poles at z,, z2,..., zm with 0< | z, | < | z21 < • • • | zm \ <R, and except for one point zQ on the boundary \ z | = R. Further, given any e>0, there must exist a neighborhood | z — z01 < 8 in which | /(z) —/(z0)| < e, provided \ z \ < R, which means that f{z) is continuous at zQ within the circle. Then at least a subsequence of[M/M] Pade approximants converges uniformly to f(z) on any compact subset of

 

 

 

 

It is regrettable that no proof yet exists. The importance of this conjecture depends on the homographie invariance property of diagonal Padé ap­proximants. The domain of pointwise convergence of a subsequence may be substantially extended by a development of the Padé conjecture.

Quasitheorem [Baker et al., 1961]. Let f(z) be analytic at z = 0, and let 6D be the union of all circles containing the origin in which f(z) is meromorphic. Then at least a subsequence of[M/M] Padé approximants converges to f(z) pointwise on any compact subset of which does not contain the poles of f(z).

"Proof.". Any point z£6D lies in the interior of a circle T with center c and radius R containing the origin O, as shown in Figure 1. Consider the mapping

 

 

 

 

and

 

The circle |w|= 1 is the image of

 

 

This is a circle for which the origin O is an interior point (if |c|</?), and O

 

Figure 2. The complex z-plane, showing the circles F, and T\.

 

and P are inverse points provided zP = — c{\R/c\2 — 1}. T is given paramet- rically by OZ=\c/R\ PZ. The result "follows" from Theorem 1.5.2.

This result justifies (7.3) and explains the choice (7.2). If /(z) is meromor- phic in T, then g( w) is meromorphic in | w | 1 and the Pade conjecture asserts that a subsequence of the diagonal sequence of Pade approximants converges to g( w) in | w | 1 (except at the poles). This quasitheorem is reinterpreted as asserting that the same subsequence of diagonal approxi­mants to /(z) converges at zeT.

Application. Consider the function f(z) = (z-a)3/2(z—b)~3/2, where a, b are arbitrary (nonzero) points in the complex plane. The Pade conjec­ture asserts the convergence, in T, of Figure 2, of a subsequence. In this example, z= oo is a regular point, and so convergence at any point outside T2 of a subsequence is asserted.

This example strongly suggests that the poles of the diagonal Pade approximants to /(z) lie on the arc of a circle through O, a, and b. In this case, the mapping t=z(a — b)/[(z — a)b] maps z=a to /= oo and z=b to t= — 1. In fact f(z) = h(t) is a Stieljtes series in t, proving that the poles of the diagonal approximants lie on the arc.

An alternative view is that the mapping u—z 1 maps the points z — a and z=b to u=a~1 and u=b~x. The exceptional set for diagonal Pade ap­proximants to f(z)=F(u) in the «-variable minimizes the capacity of the cut, and therefore is the straight line from a-1 to b~x. This line is the image of the arc of the circle through z = 0, z = a, and z=b from a to b.

The status of the Pade conjecture and its relation to open problems in rational approximation is surveyed by Walsh [1970].