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.
