Abstract
While the theory of asymptotics for L2-minimal polynomials is highly developed, so far not much is known about Lp-minimal polynomials, \(p\in (1,\infty) \backslash \{2\}.\) Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form \(w(x)/\sqrt{1-x^2},\) where w is positive and \(w' \in {\rm Lip}\,\alpha\) on [-1,1] with \(\alpha \in (0,1)\ {\rm if}\ p\geq 2\) and \(\alpha > (2/p) - 1\ {\rm if}\ 1<p\leq 2.\)
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Kroo, A., Peherstorfer, F. Asymptotic Representation of Lp-Minimal Polynomials, 1 < p < ∞. Constr Approx 25, 29–39 (2007). https://doi.org/10.1007/s00365-006-0628-5
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DOI: https://doi.org/10.1007/s00365-006-0628-5