Abstract
We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] 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 [Formula: see text]th root asymptotic behavior of the corresponding sequence of Sobolev extremal polynomials and their derivatives.
Highlights
Let μ0 be a positive Borel measure supported on an interval of the real line Δ0
We denote by Pn the space of polynomials of degree ≤ n, and by P∗n ⊂ Pn the subset of monic polynomials of exact degree n
For j = 0 the zeros of the polynomials Ln can abandon Δ and their asymptotic zero distribution is governed by the balayage of μΔ onto a certain region which we describe later
Summary
Let μ0 be a positive Borel measure supported on an interval of the real line Δ0 (which does not reduce to a point). When p = 2, and the norm (6) is given by an inner product, the corresponding Sobolev extremal polynomials (or orthogonal with respect to the associated inner product) have been extensively studied. To the study of the asymptotic distribution of the zeros and critical points of the Sobolev extremal polynomials. Let μ be a finite Borel measure whose compact support S(μ) ⊂ C has positive logarithmic capacity and let Pn be the associated monic orthogonal polynomial with respect to μ of degree n.
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