Approximaths

Padé approximants: Convergence I

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For row sequences on the Padé table, Montessus’s theorem (1902) proves convergence for functions meromorphic on a disk. Before giving the statement of the theorem, we would like to remind the reader of a few definitions:

Holomorphic function: A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of each point in a given domain.

Analytic function: An analytic function is a function that is locally given by a convergent power series.

Meromorphic function: A meromorphic function on an open subset D of the complex \mathbb{C}-plane is a function that is holomorphic on all of D except for a set of isolated points. These points are called the ‘poles’ of the function.

Here is the Montessus’s theorem as stated by E. B. Saff in 1972:

Let f(z) be analytic at z = 0 and meromorphic with precisely \nu poles (multiplicity counted) in the disk |z| < \tau. Let D be the domain obtained from |z| < \tau by deleting the \nu poles of f(z). Then, for all m sufficiently large, there exists a unique rational function R_{m, \nu} of type (m, \nu), which interpolates f(z) in the point z = 0 considered of multiplicity m+\nu+1. Each R_{m, \nu} has precisely \nu finite poles and, as m \to \infty, these poles approach the \nu poles of f(z) in |z| < \tau. The sequence R_{m, \nu} converges in D to f(z), uniformly on any compact subset of D.

Alternative formulation:

Let f(z) be meromorphic in |z| < \tau, analytic at z = 0, and with a total of \nu poles \zeta_1, \zeta_2, \dots, \zeta_{\nu} (with multiplicity included) in |z| < \tau. Then, as m \to \infty, the Padé approximants P(m,\nu) of f converge on:

\displaystyle S_f := \{ z \in \mathbb{C} \mid |z| < \tau \} \setminus \{ \zeta_1, \zeta_2, \dots, \zeta_\nu \}

to f, uniformly on every compact subset K of S_f. In particular:

\displaystyle P(m,\nu)(z) \to f(z)

The Padé table is represented as follows:

n=0 n=1 n=2 \dots n=\nu \dots
m=0 P(0,0) P(0,1) P(0,2) \dots \boxed{P(0,\nu)} \dots
m=1 P(1,0) P(1,1) P(1,2) \dots \boxed{P(1,\nu)} \dots
m=2 P(2,0) P(2,1) P(2,2) \dots \boxed{P(2,\nu)} \dots
\dots \dots \dots \dots \dots
m \to \infty \dots \dots \dots \boxed{P(m,\nu)} \dots

The Montessus’s theorem is crucial in approximation theory as it ensures the uniform convergence of Padé approximants for meromorphic functions, enhancing the accuracy of rational approximations.

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