Multipoint Padé Approximation and Orthogonal Rational Functions

Abstract

Let {a₁,...,a_p} be fixed points in the complex plane C, and denote by R the space of all functions of the form R(z) = ${\alpha _0} + \sum\limits_{i = 1}^p {\sum\limits_{j = 1}^{{N_i}} {\frac{{{\alpha _{ij}}}}{{{{\left( {z - {a_i}} \right)}^j}}},{\alpha _0},{\alpha _{ij}} \in C} }$. Let the series $\frac{{{}^C0}}{z}$ and $\sum\limits_{j = 1}^\infty {c_j^{(i)}{{\left( {z - {a_i}} \right)}^{j - 1}},i = 1,2,...,p,}$ be given, and define the linear functional Φ on R by Φ((z-a_i)^j) = c⁽ⁱ⁾ _j, j=1,2,..., i=1,...,p, Φ(1) = c₀.. Define the bilinear form <, > on RxR by <A,B> = Φ(A.B), and assume that Φ(R²) ≠ 0 when R(t) ≢ 0. The Gram-Schmidt process applied to the sequence $\left\{ {1,\frac{1}{{\left( {z - {a_1}} \right)}},...,\frac{1}{{\left( {z - {a_p}} \right)}},\frac{1}{{{{\left( {z - {a_1}} \right)}^2}}},...} \right\}$ gives an orthogonal system {Q_n (z)} of functions in R. These orthogonal functions can be used to construct multipoint Padé approximants for the given series.