Asymptotics of columns in the table of orthogonal polynomials with varying measures (Q1907528)

Let \(\{\mu_k\}\) be a given sequence of finite positive Borel measures with compact supports and define for \(p> 0\) and \(n= 0, 1,\dots\) the polynomial \(P_{n, p}(z; \mu_k)= z^n+\cdots\) to be an \(L_p(\mu_k)\) extremal polynomial of degree \(n\). The author considers columns \(k= 1, 2,\dots\) in the table of ``orthogonal polynomials \(P_{n, p}(z; \mu_k)\) (\(n\) fixed) with varying measure'' (actually only for \(p= 2\) the polynomials are orthogonal and for \(0< p\leq 1\) they might very well be not unique). First, he proves several results in the situation that \(\mu_k\to u (*)\) (i.e., convergence in the weak star topology on the set of all finite positive Borel measures w.r.t. continuous functions with compact support) and then applies his general results to the special cases, where the measures are supported on the unit circle (where do the zeros of largest modulus converge to and including the treatment of the location of so-called \textit{uninteresting zeros}), supported on the real line (showing that the converse of main Theorem 2.1 does not hold in general); moreover, he extends his methods to measures whose supports belong to the real line and are not compact. A well written and interesting paper.