The convergence of Padé approximants of meromorphic functions

Convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on the real line

We obtain results on the convergence of Padé approximants of Stieltjes-type meromorphic functions and the relative asymptotics of orthogonal polynomials on unbounded intervals. These theorems extend some results given by Guillermo López in this direction substituting the Carleman condition in his theorems by the determination of the corresponding moment problem. c © 2009 Elsevier Inc. All right...

Convergence in Capacity of Rational Approximants of Meromorphic Functions

Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy Eρ(f), ρ(f) < ∞. We investigate rational approximants with numerator degree 6 n and denominator degree 6 mn for f . We show that the geometric convergence rate on E implies convergence in capacity outside E if mn = o(n) as n → ∞. Further, we show that the condition is sharp and that the convergence in capacity...

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...