Nevanlinna parametrizations for the extended interpolation problem

Sechiko Takahashi

Vol. 146 (1990), No. 1, 115–129

DOI: 10.2140/pjm.1990.146.115

Abstract

Let ℬ be the set of holomorphic functions f with |f|≤ 1 in the open unit disc D = {z ∈ ℂ : |z| < 1}. Let σ = {z₁,z₂,…} be a finite or infinite sequence of distinct points in D and, for each point z_i ∈ σ, (c_i0,…,c_ini−1) be an ordered n_i-tuple of complex numbers (0 < n_i < +∞). The problem is to find a function f which belongs to ℬ and satisfies the extended interpolation conditions

f(z) = ∑ α=0ni−1c iα(z − zi)α + O((z − z i)ni ) (∀zi ∈ σ).

(E1)

Let ℰ denote the set of all solutions of this problem (E1) in ℬ and assume the hypothesis

ℰ has at least two elements.

(H)

A bijection π : ℬ→ℰ is called Nevanlinna parametrization of ℰ if there exist four functions P, Q, R, and S holomorphic in D and such that Rg + S≢0, π(g) = (Pg + Q)∕(Rg + S) for any g ∈ℬ. The existence, some properties and some applications of such parametrizations are shown. One has a bijection between the set of Nevanlinna parametrizations of ℰ and the group of Möbius transformations.

Mathematical Subject Classification 2000

Primary: 30E05

Milestones

Received: 18 November 1988

Revised: 1 August 1989

Published: 1 November 1990