Weighted polynomial approximation for convex external fields
In this landmark paper, Vilmos Totik largely settles one of the problems that has intrigued researchers in orthogonal polynomials, extremal polynomials, exponential weights, \(\dots\). Let \(\Sigma\subset\mathbb{R}\) and \(Q: \Sigma \to\mathbb{R}\). What functions may be uniformly approximated on \(\Sigma\) by weighted polynomials of the form \(P_n\exp (-nQ)\) as \(n\to\infty\)? In this question, potential theory for external fields plays a crucial role. It is known that there is a certain equilibrium measure \(\mu_Q\) associated with \(Q\), and with support \(S_Q\) in \(\Sigma\), that governs the solution of this problem. If \(f:\sum \to\mathbb{R}\) and there exist polynomials \(P_n\) of degree \(\geq n\) such that \[ \lim_{n\to \infty}\bigl \|f-P_n\exp(-nQ)\bigr \|_{L_\infty (\Sigma)}=0, \] then \(f\) must vanish outside \(S_Q\). The author essentially settles the converse for convex \(Q\): if \((a,b)\) is contained in \(S_Q\) and \(Q\) is convex on \((a,b)\), then every function \(f\) vanishing outside \((a,b)\) may be uniformly approximated on \(\Sigma\) by weighted polynomials of the form \(P_n\exp (-nQ)\). This follows from more general results about equilibrium measures with ``smooth integral''. The methods of proof confirm that Totik is a technical virtuoso; he uses his method of very careful discretisation of a potential in a manner that only he could have invented. The paper is essential reading for those involved in potential theory for external fields, weighted approximation, orthogonal polynomials for exponential weights,\(\dots\).
scientific article; zbMATH DE number 1471468