# Necessary and Sufficient Conditions - Nehari

I want to find necessary and sufficient conditions in order that the following problem has a solution:

Given complex numbers $a_0,\ldots,a_n$ consider the problem to find all complex function $f$, $$f(\lambda)=\sum_{\nu=0}^{\infty}{\lambda^\nu f_\nu},$$

satisfying the following three conditions:

• $f_j=a_j$ for $j=0,\ldots,n$;
• $\sum_{\nu=0}^{\infty}|f_\nu|<\infty$;
• $\sup_{\lambda\in \mathbb{D}}|f(\lambda)|<1,$ where $\mathbb{D}$ is the open unit disk in the complex plane.

First, I had no idea how to start, however after some research I found out that this problem looks similar to a Nehari extension problem. see: https://www.encyclopediaofmath.org/index.php/Nehari_extension_problem

Apparently, a lot is known about the Nehari extension problem. So, I have tried to reduce my problem to a Nehari EP to use the existence theorems, but have had no success. How can I transform this system into a Nehari extension problem?

Any hints are appreciated.

• Do you mean $\sup_\lambda |f(\lambda)|$? Supremum over what set? – Robert Israel Sep 13 '16 at 20:52
• @RobertIsrael Sorry. See edit. – KayL Sep 13 '16 at 21:00