On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions

TL;DR

This paper constructs a specific analytic function with branch points outside an interval, demonstrating that in certain potential-theoretic problems, no minimal compact set exists for rational approximation.

Contribution

It provides a counterexample showing the non-existence of minimal compacta in a particular extremal potential problem for rational approximation.

Findings

No minimizing compacta exist for the constructed function.

The function has four second-order branch points outside the interval.

The problem relates to extremal potential theory in rational approximation.

Abstract

For an interval $E=[a,b]$ on the real line, let $\mu$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^{\mu}$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the family of the admissible compacta of $f$ has no minimizing elements with regard to the extremal theoretic-potential problem, in the external field equals $V^{-\mu}$.