Non-uniqueness of rational best approximants
Let \(V\) be a complement in the extended complex plane \(\widetilde \mathbb{C}\) of the closed unit disc, and let \(H^2(V)\) be the associated Hardy space with the norm \[ \|g\|=\lim_{r\to 1+} \left({1\over 2\pi} \int^{2\pi}_0 \bigl|g(re^{it}) \bigr|^2 dt\right)^{1/2}. \] \({\mathcal R}_{m,n}\) denotes the set of rational functions with numerators and denominators \(\leq m,n\) respectively; \({\mathcal R}^1_{m,n}\) is the subset of \({\mathcal R}_{m,n}\) consisting of functions all of whose poles lie in the open unit disc; thus \({\mathcal R}^1_{m,n}= {\mathcal R}_{m,n}\cap H^2(V)\). A function \(f\) of the form \(f(z)= \int(t-z)^{-1} d\mu(t)\), where \(\mu\) is a positive measure of compact support in \(\mathbb{R}\) is a Markov function; if \(\text{supp} \mu \subseteq(-1,1)\) then \(f\in H^2(V)\). The paper is concerned with the uniqueness or non-uniqueness of best approximation to Markov functions \(f\in H^2(V)\) by functions of \({\mathcal R}^1_{n-1,n}\) for \(n\in\mathbb{N}\), and it presents two main results. Theorem 4: There exist Borel measures \(\mu\) with countable \(\text{supp} \mu \subseteq(-1,1)\) such that for each odd \(n\in\mathbb{N}\) the best approximation in \(H^2(V)\) to \(f\) from \({\mathcal R}^1_{n-1, n}\) is not unique. Theorem 5: Let \(n_0\in \mathbb{N}\) be odd. There exists a Borel measure \(\mu\) such that \(\text{supp} \mu\) is a closed subinterval of \((-1,1)\), \(\mu\) has a positive smooth density function on \(\text{supp} \mu\), and for each odd \(n\leq n_0\) the best approximation in \(H^2(V)\) to \(f\) from \({\mathcal R}^1_{n-1,n}\) is not unique. The measure \(\mu\) constructed for Theorem 5 belongs to the Szegö class and it follows from a result of the same authors (Asymptotic uniqueness of best rational approximants of given degree to Markov functions in \({\mathcal L}^2\) of the circle (to appear) stated here as Theorem 3, that best approximants to \(f\) from \({\mathcal R}^1_{n-1,n}\) are unique for all but a finite set of \(n\in\mathbb{N}\).
scientific article; zbMATH DE number 1334857