Generalized moment representations and multipoint Padé approximants

Abstract

By using the method of generalized moment representations, we develop an approach to the construction and investigation of multipoint Padé approximants.Along with the classical Padé approximants, the so-called multipoint Padé approximants provide one of the most important tools for rational approximation of functions. Definition 1 [1, p. 289]. Let z 1 , z 2 , … , z R ∈ Ᏸ be points belonging to a connected subset Ᏸ of the complex plane, let M and N be nonnegative integer numbers, and let L = ( l 1 , l 2 , … , l R ) be a vector with positive integer coordinates such that l r r R = ∑ 1 = M + N + 1. We say that a rational function [ ] ( … ) = ( ) ( ) M N z z z P z Q z f L R M N / , , ; 1 , (1) where P M ( z ) and Q N ( z ) are algebraic polynomials of degrees ≤ M and ≤ N, respectively, is a multipoint (or R-point) Padé approximant of order [ M / N ] at the points z 1 , z 2 , … , z R of index L for a function f analytic in the domain Ᏸ ifIt is obvious that, for R = 1 and z 1 = 0, this definition coincides with the definition of classical Padé approximants (see [1, p. 292]). In different publications, multipoint Padé approximants are also called rational interpolants, Newton -Padé approximants, etc.An approach to the construction and investigation of two-point Padé approximants is described in [2]. It is based on the application of generalized moment representations introduced by Dzyadyk in [3]. Below, we give the corresponding definition.

Approximation of the inverse Langevin function revisited

“…According to Baker and Graves-Morris (1981), we can find in the literature a few names for N-point Padé approximants like multipoint Padé approximants (Golub 2003(Golub , 2004, rational interpolants, or Newton Padé approximants. Despite the wealth of research on different approximations …”