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Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality. (English) Zbl 1155.33006

The authors “study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form \(e^{-\varphi(x)}\), giving a unified treatment for the so-called Freud (i.e., when \(\varphi\) has polynomial growth at infinity) and Erdös (when \(\varphi\) grows faster than any polynomial at infinity) cases”.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A10 Approximation by polynomials
30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:

[1] Duran, A.; Saff, E., Zero location for nonstandard orthogonal polynomials, J. Approx. Theory, 113, 127-141 (2001) · Zbl 1011.42013
[2] Erdös, P., On the distribution of the roots of orthogonal polynomials, (Alexits, G.; etal., Proc. Conf. Constr. Th. Fns. (1972), Academiai Kiado: Academiai Kiado Budapest), 145-150
[3] Gautschi, W.; Kuijlaars, A. B.J., Zeros and critical points of Sobolev orthogonal polynomials, J. Approx. Theory, 91, 117-137 (1997) · Zbl 0897.42014
[4] Gonchar, A. A.; Rakhmanov, E. A., Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. USSR Sb., 53, 119-130 (1986) · Zbl 0618.30008
[5] Knopfmacher, A.; Lubinsky, D. S., Analogues of Freud’s conjecture for Erdös type weights and related polynomial approximation problems, (Lecture Notes in Math., vol. 1287 (1987), Springer-Verlag: Springer-Verlag Berlin), 21-69 · Zbl 0638.41009
[6] Levin, E.; Lubinsky, D. S., Orthogonal Polynomials for Exponential Weights, CMS Books Math., vol. 4 (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0997.42011
[7] López Lagomasino, G.; Marcellán, F.; Pijeira, H., Logarithmic asymptotics of contracted Sobolev extremal polynomials on the real line, J. Approx. Theory, 143, 62-73 (2006) · Zbl 1106.41031
[8] Mhaskar, H. N.; Saff, E., The distribution of zeros of asymptotically extremal polynomials, J. Approx. Theory, 65, 279-300 (1991) · Zbl 0744.41007
[9] Rakhmanov, E. A., On asymptotic properties of polynomials orthogonal on the real axis, Mat. USSR Sb., 47, 155-193 (1984) · Zbl 0522.42018
[10] Saff, E.; Totik, V., Logarithmic Potentials with External Fields, Grundlehren Math. Wiss., A Series of Comprehensive Studies in Mathematics, vol. 316 (1997), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0881.31001
[11] Totik, V., Weighted Approximation with Varying Weights, Lecture Notes in Math., vol. 1569 (1994), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York
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