Convergence of two-point Padé approximants to piecewise holomorphic functions

Convergence of Two-point Padé Approximants to Piecewise Holomorphic Functions

Let f0 and f∞ be formal power series at the origin and infinity, and Pn/Qn, with deg(Pn), deg(Qn) 6 n, be a rational function that simultaneously interpolates f0 at the origin with order n and f∞ at infinity with order n + 1. When germs f0, f∞ represent multi-valued functions with finitely many branch points, it was shown by Buslaev [5] that there exists a unique compact set F in the complement...

Convergence of Multipoint Padé-type Approximants

Let µ be a finite positive Borel measure whose support is a compact subset K of the real line and let I be the convex hull of K. Let r denote a rational function with real coefficients whose poles lie in C \ I and r(∞) = 0. We consider multipoint rational interpolants of the function f (z) = dµ(x) z − x + r(z), where some poles are fixed and others are left free. We show that if the interpolati...

Shanks' convergence acceleration transform, Padé approximants and partitions

General order multivariate Padé approximants for Pseudo-multivariate functions. II

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E (x) = ∞ ∑ j1,j2,...,jm=0 x1 1 x j2 2 · · ·x jm m (j1 + j2 + · · ·+ jm)! , the logarithm function L(x) = ∑ j1+j2+···+jm≥1 x1 1 x j2 2 · · ·x jm m j1 + j2 + · · ·+ jm , the Lauricella funct...

General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inheri...