Analysis Seminar: On Rational Approximants of Multi-Valued Functions

Abstract: Ever since the work of Runge in the late 19th century, it is known that functions analytic in a neighborhood of a compact set can be approximated arbitrarily close by rational functions (later Vitushkin characterized the compacta on which such an approximation is possible.) Early in the 20th century, Walsh has shown that

\[
\limsup_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A \leq \inf_B \exp\left\{ - 1/\mathrm{cap}(A,B) \right\},
\]
where \( f \) is holomorphic in a neighborhood of a continuum \( A \), \( \mathcal R_n \) is the set of rational functions of type \( (n,n) \), \( \mathrm{cap}(A,B) \) is the condenser capacity, and the infimum on the right is taken over all compact sets \( B \) such that \( f \) is holomorphic in the complement of \( B \) (the complement must be connected and necessarily contain \( A \)).  In general this bound is sharp. Driven by evidence from certain classes of functions, Gonchar has conjectured that
\[
\liminf_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A \leq \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\}.
\]
This conjecture was shown to be true by Parfenov with the help of Adamyan-Arov-Krein approximants. Elaborating on the work of Stahl, Gonchar and Rakhmanov have shown that
\[
\lim_{n\to\infty} \inf_{r\in\mathcal R_n} \|f-r\|_A = \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\}
\]
if \( f \) is a multi-valued function meromorphic outside of a compact polar set. For a subclass of such functions, asymptotic distribution of poles of sequences of rational approximants  \( \{ r_n \} \) such that
\[
\lim_{n\to\infty} \|f-r_n\|_A = \inf_B \exp\left\{ - 2/\mathrm{cap}(A,B) \right\},
\]
where \( A \) is a continuum, will be discussed.

Host: Brett Wick