Rational best approximants in the Hardy space \(H^2\) (Q2781667)

In this survey paper the author presents (without proofs) recent developments in the study of rational best approximation of Markov functions in the norm of the Hardy space \(H^2(V)\) with \(V=\{z\in {\overline{\mathbb{C}}}:|z|>1\}.\) The author has a major contribution in this field. We recall that a Markov function \(f\) with respect to positive measure \(\mu,\) \((supp(\mu)\subseteq (-1,1))\) is represented by \(f(z)=\int(t-z)^{-1}d\mu(t).\) The analysis of rational best approximants in \(H^2(V)\) requires some results about orthogonal polynomials with varying weights and of multipoint Padé approximants. Then, asymptotic error estimates in the weak and strong sense for multipoint Padé approximants are presented at the beginning. NEWLINENEWLINENEWLINELet \(\mathcal{R}^1_{n-1,n}\) be the family of all rational functions in \(H^2(V),\) of the form \(p_{n-1}/p_n,\) with poles in the unit disk, \(p_k\) being a polynomial of degree \(k.\) Let \(R^*_n\in {\mathcal R}^1_{n-1,n}\) be a best approximant of the Markov function \(f.\) The author gives error estimates of \(\|f-R^*_n\|^{1/n}_{H^2}\) and \(\|f-R^*_n\|_{H^2},\) as well as the point-wise variants, for \(n\rightarrow\infty.\) The estimates are expressed in terms of the condenser capacity for a pair of disjoint compact sets. The uniqueness of \(R^*_n\) is obtained for large \(n.\)NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].