The Padé table of functions having a finite number of essential singularities

Albert Edrei

Vol. 56 (1975), No. 2, 429–453

DOI: 10.2140/pjm.1975.56.429

Abstract

Let f(z) be regular at the origin and let it be single-valued and regular except for poles and s + 1 < +∞ essential singularities α₀,α₁,α₂,⋯,α_σ. The α’s may be limit-points of poles and α₀ = ∞ is permissible. Assume that α_k is of finite order λ_k and let Λ = ∑ λ_k.

The author obtains a convergence theorem for the Padé table of ∑ a_jz^j = f(z). The simplest consequence of his result may be stated as follows: if Λ < 2, the diagonal approximants of the Padé table converge almost everywhere in the plane.

Mathematical Subject Classification

Primary: 30A82

Milestones

Received: 6 November 1973

Published: 1 February 1975