The Multivariate Herglotz-Nevanlinna Class: Superresolution

Unique solutions of bounded, or equivalently, non-negative real part, holomorphic interpolation in several complex variables with finite prescribed data are very rare, with rational inner functions in a polydisk as the best understood examples. We analyze the continuity of global solutions as functions of the finite interpolation data in neighborhoods of elements distinguished by this uniqueness property. Our study covers rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.