Rational inner interpolation in dimensions d > 2

Determine whether, for any dimension d > 2, any finite set of points z_1, …, z_N in the polydisk D^d, and any analytic function f: D^d → D, there exists a d-variable rational inner function φ such that φ(z_j) = f(z_j) for all j = 1, …, N.

Background

In two variables, Agler’s Pick interpolation theorem characterizes solvability via positive semidefinite kernels and rational inner transfer-function realizations. The paper notes that, in higher dimensions, even the existence of an analogous finite-dimensional reduction for interpolation is unknown. A natural weaker question, stated here as Problem ratinterp, asks whether rational inner functions can interpolate finite sets of values of bounded analytic maps in dimensions d > 2.

The authors remark that this question has an affirmative answer for N ≤ 3 by prior work, but the general case remains unsettled, highlighting a fundamental gap in multivariable interpolation theory beyond the bidisk.