Meromorphic Extendibility and Rigidity of Interpolation

TL;DR

This paper characterizes when a Holder continuous function on the unit circle extends meromorphically into the disk with limited poles, based on winding number conditions involving algebraic perturbations.

Contribution

It provides a necessary and sufficient condition for meromorphic extendibility with bounded poles using winding number criteria and algebraic approximation.

Findings

Characterizes meromorphic extendibility via winding number conditions.

Establishes a link between Holder continuity and meromorphic extension.

Provides a criterion for the maximum number of poles in extension.

Abstract

Let T be the unit circle, f be an \alpha-Holder continuous function on T, \alpha>1/2, and A be the algebra of continuous function in the closed unit disk \bar D that are holomorphic in D. Then f extends to a meromorphic function in D with at most m poles if and only if the winding number of f+h on T is bigger or equal to -m for any h\in A such that f+h \neq 0 on T.