Recursive interpolating sequences

Abstract

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(z 1) = w 1 and f(z n+1) = an f(zn ) + w n+1. We add a recursion for the derivative of the type: f′(z 1) = w 1 ′ $\begin{array}{} w_1' \end{array} $ and f′(z n+1) = a n ′ $\begin{array}{} a_n' \end{array} $ [(1 − |zn |2)/(1 − |z n+1|2)] f′(zn ) + w n + 1 ′ , $\begin{array}{} w_{n+1}', \end{array} $ where ( a n ′ $\begin{array}{} a_n' \end{array} $ ) is bounded and ( w n ′ $\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.