Integrability and regularity of rational functions

TL;DR

This paper investigates the integrability and boundary regularity of rational functions on the bidisk and polydisk, providing algebraic characterizations, explicit generators, and dimension counts, with implications for non-tangential limits and boundedness.

Contribution

It offers a new algebraic framework for understanding integrability and regularity of rational functions on the bidisk, including explicit generators and dimension formulas.

Findings

Rational inner functions on the polydisk have non-tangential limits at every point of the n-torus.

Explicit generators for the ideal of square-integrable rational functions are provided.

Rational functions square integrable on the two-torus are non-tangentially bounded at all points.

Abstract

Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting contractions on a finite dimensional Hilbert space and studying their joint generalized eigenspaces. Non-tangential regularity of rational functions on the polydisk is also studied. One result states that rational inner functions on the polydisk have non-tangential limits at every point of the n-torus. An algebraic characterization of higher non-tangential regularity is given. We also make some connections with the earlier material and prove that rational functions on the bidisk which are square integrable on the two-torus are non-tangentially bounded at every point. Several examples are provided.