Sobolev extremal polynomials with respect to mutually singular measures

TL;DR

This paper investigates Sobolev extremal polynomials with respect to mutually singular measures, analyzing their critical points, distribution, and asymptotic behavior, revealing new insights into their structure and properties.

Contribution

It establishes the simplicity and location of critical points for Sobolev extremal polynomials with mutually singular measures and studies their asymptotic distributions and root behavior.

Findings

Critical points are simple and lie within the convex hull of the measures' support.

Asymptotic distribution of critical points is characterized.

The nth root asymptotics of the polynomials and their derivatives are derived.

Abstract

We consider extremal polynomials with respect to a Sobolev-type $p$-norm, with $1<p<\infty$ and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and contained in the interior of the convex hull of the support of the measures involved and the asymptotic critical point distribution is studied. We also find the $n$th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.