Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Abstract

Let $E\subset\mathbb{C}$ be a compact set with connected complement, and let $\rho_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on $E$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively. The Saff--Gonchar theorem is the fundamental result describing the asymptotic behavior of rational approximation errors along the rows of the Walsh table. It was first proved by Saff for continua with connected complement and Jordan boundary and subsequently extended by Gonchar to regular compact sets. Motivated by a comparison of the Saff--Gonchar theorem with Hadamard's classical theorem on Hankel determinants, we study, for each fixed $m\ge 0$, the asymptotic behavior as $n\to\infty$ of the products $$ \prod_{k=0}^{m}\rho_{n-m+k,k}(f;E). $$ We establish Hadamard-type asymptotic formulas for these products on the closed unit disc and, more generally, on continua with connected complement and Jordan boundary. In the disc case, our approach combines Hadamard's classical theorem and the Saff--Gonchar theorem with weighted Hankel operators and an AAK-type theorem for meromorphic approximation. We also show that there exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets $E_R$ is attained.