Pick interpolation in several variables

TL;DR

This paper explores the Pick interpolation problem in multiple complex variables, specifically in polydisks and unit balls, employing dual algebra methods and factorization results to extend Pick theorems to bounded regions in complex space.

Contribution

It introduces new factorization techniques for Bergman spaces and generalizes Pick theorems to higher-dimensional bounded regions in complex analysis.

Findings

Established a Pick theorem for bounded regions in d

Developed dual algebra techniques for multivariable Pick problems

Extended Pick interpolation results to complex polydisks and balls

Abstract

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb{C}^d$.