Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Let ${\rho }_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on a compact set $E\subset \mathbb{C}$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively. Motivated by Hadamard’s classical theorem on Hankel determinants and by Gonchar’s theorem on rows of the Walsh table, we study, for each fixed $m\ge 0$, the asymptotic behavior as $n\to \mathrm{\infty }$ of the products $$\prod _{k=0}^m{\rho }_{n-m+k,k}(f;E)\mathrm{.}$$ 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 Gonchar’s 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.