Recursive interpolating sequences

Abstract

This paper is devoted to pose several interpolation problems on the open unit disk 𝔻 of the complex plane in a recursive and linear way. We look for interpolating sequences (zn) in 𝔻 so that given a bounded sequence (an) and a suitable sequence (wn), there is a bounded analytic function f on 𝔻 such that f(z1) = w1 and f(zn+1) = anf(zn) + wn+1. We add a recursion for the derivative of the type: f′(z1) = w1′$\begin{array}{} w_1' \end{array} $ and f′(zn+1) = an′$\begin{array}{} a_n' \end{array} $ [(1 − |zn|2)/(1 − |zn+1|2)] f′(zn) + wn+1′,$\begin{array}{} w_{n+1}', \end{array} $ where (an′$\begin{array}{} a_n' \end{array} $) is bounded and (wn′$\begin{array}{} w_n' \end{array} $) is an appropriate sequence, and we also look for zero-sequences verifying the recursion for f′. The conditions on these interpolating sequences involve the Blaschke product with zeros at their points, one of them being the uniform separation condition.

Keywords

• interpolating sequence
• uniformly separated sequence
• bounded analytic function
• 30e05
• 30h05
• 30j10