A uniqueness theorem for bounded analytic functions on the polydisc

TL;DR

This paper establishes a set of points in the polydisc such that any analytic function matching a rational inner function of bounded degree at these points must coincide with it everywhere, ensuring uniqueness in the Pick problem.

Contribution

The paper constructs explicit point sets in the polydisc that guarantee uniqueness of bounded analytic functions matching rational inner functions of bounded degree.

Findings

Unique determination of rational inner functions by finite data points.

Explicit construction of point sets for the Pick problem in the polydisc.

Ensures uniqueness of solutions for bounded analytic functions with prescribed data.

Abstract

For each n,N>0 we construct a set of points x_1,...,x_M in D^n with the following property: if f is a rational inner function on D^n of degree strictly less than N and g is an analytic function mapping D^n to D that satisfies g(x_i)=f(x_i) for each i=1,...,M, then g=f on D^n. In terms of the Pick problem on D^n, our result implies that for any rational inner f of degree less than N, the Pick problem with data x_1,...,x_M and f(x_1),...,f(x_M) has a unique solution.