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Asymptotic zero distribution for a class of extremal polynomials. (English) Zbl 1465.42001

Summary: 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.

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C05 Real polynomials: analytic properties, etc.
26C10 Real polynomials: location of zeros
33C47 Other special orthogonal polynomials and functions
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