Boundary interpolation by finite Blaschke products

TL;DR

This paper characterizes all finite Blaschke products of degree at most n-1 that interpolate given boundary points on the unit circle with specified target values, including cases where the maximal degree is minimal.

Contribution

It provides a complete description of boundary interpolation by finite Blaschke products and identifies conditions for minimal degree solutions.

Findings

Explicit formulas for all interpolating Blaschke products of degree ≤ n-1.

Characterization of cases where degree n-1 is the minimal possible.

Conditions under which solutions exist and are unique.

Abstract

Given $n$ distinct points $t_1,\ldots,t_n$ on the unit circle $\T$ and equally many target values $\f_1,\ldots,\f_n\in\T$, we describe all Blaschke products $f$ of degree at most $n-1$ such that $f(t_i)=\f_i$ for $i=1,\ldots,n$. We also describe the cases where degree $n-1$ is the minimal possible.