# Recursive interpolating sequences

Recursive interpolating sequences AbstractThis 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) = an f(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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

# Recursive interpolating sequences

, Volume 16 (1): 8 – Apr 30, 2018
8 pages

/lp/degruyter/recursive-interpolating-sequences-8lScE6H09w
Publisher
de Gruyter
ISSN
2391-5455
eISSN
2391-5455
D.O.I.
10.1515/math-2018-0044
Publisher site
See Article on Publisher Site

### Abstract

AbstractThis 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) = an f(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.

### Journal

Open Mathematicsde Gruyter

Published: Apr 30, 2018

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