# 1.5 Duality and Invariance

Theorem 1.5.1 (Duality).

In this section, we state some algebraic properties of Padé approximants which are attractive features of a class of approximating functions. The theorems are easy to prove, and the interpretation of the theorems is important. All the theorems of the section concern algebraic properties of power series, and no convergence property is in any way directly implied.

The duality may be summarized by saying that the Padé approximant of the reciprocal function is the reciprocal of the Padé approximant of the function; it may be glibly restated by saying that Padé approximation and reciprocation commute.

If a class of functions and the reciprocal functions have a common property, e.g. they are meromorphic, then the duality property shows a valuable symmetry feature of the Padé approximants as approximating functions.

Theorem 1.5.2 (Homographie invariance under argument transformations). Let of the argument

Define an origin preserving bilinear transformation

Notice that the proof is only valid for diagonal approximants, and so this homographie invariance only holds for the diagonal sequences. Theorem 1.5.2 is usually called the theorem of Baker, Gammel, and Wills [1961]; a closely related result was previously proved by Edrei [1939].

The homographie invariance theorem for argument transformations is the cornerstone of the optimistic approach to Padé approximants. This optimism is entirely validated by practical experience, but not as yet by proven theorems. The fundamental reason for this optimism is that the mapping shown in Figure 1 allows any circle T in the z-plane | w\ = p centered on the origin. If a sequence of diagonal approximants may be proved to converge to within then convergence of the same

sequence for /(z) follows in the interior of T (see the quasitheorem of Section 6.7). We also understand the acceleration of convergence using the sequence of diagonal Padé approximants, mentioned in Chapter 3, in terms of a generalized Euler transformation [Thacher, 1974]. Other implications of this theorem are discussed in Section 6.7.

enclosing the origin as an interior point to be mapped onto a given circle a function we define

Homographie invariance of the values, like invariance under argument transformations, is generally only valid for diagonal approximants. An interesting feature of this result is that the value oo of a Padé approximant is treated on a par with any other value; this is significant in the context of convergence on the Riemann sphere (see Section 6.4). The transformation

can be broken down into successive elementary trans formations, each with a simple interpretation: the mapping is a composite of translations and inversion given.

Details of the interpretation are to be found in EPA, p. 113.

This theorem is used repeatedly in Chapter 5, where we prove a series of results for [M— 1 /M] Pade approximants and generalize them to [M+J/M] Pade approximants for — 1, using a method equivalent to the truncation theorem.

This theorem is summarized by saying that diagonal Padé approximants preserve unitarity. It is important in S-matrix theory (see Part II, Chapter 4) because the S-matrix is unitary. A timely note of caution is that complex conjugation may destroy analyticity, and it is prudent to define

The invariance theorems of this section have justified the value of Pade approximants as practical approximating functions. In fact, one obvious esthetic test of the merit of any generalization of Padé approximants is whether the generalizations have such useful invariance properties.

Exercise 1 Let f(z) satisfy the conditions of Theorem 1.5.5. Define so that t{z) obeys a unitarity condition