A Numerical Model for the Construction of Finite Blaschke Products with Preassigned Distinct Critical Points

Abstract

We present a numerical model for determining a finite Blaschke product of degree $n+1$ having $n$ preassigned distinct critical points $z_1,\dots,z_n$ in the complex (open) unit disk $\mathbb{D}$. The Blaschke product is uniquely determined up to postcomposition with conformal automorphisms of $\mathbb{D}$. The proposed method is based on the construction of a sparse nonlinear system where the data dependency is isolated to two vectors and on a certain transformation of the critical points. The efficiency and accuracy of the method is illustrated in several examples.