Full quadrature sums for pth powers of polynomials with Freud weights |
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Authors: | D S Lubinsky D M Matjila |
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Institution: | a Department of Mathematics, University of the Witwatersrand, P.O. Wits 2050, 1 Jan Smuts Ave, Private Bag 3, Johannesburg, South Africa b Department of Mathematics, University of the North, Private Bag X1106, Sovenga 0727, South Africa |
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Abstract: | In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate 1 p < ∞, for every polynomial P of degree at most n + rn1/3, where W2 is a Freud weight such as exp(−¦x¦), > 1, λjn are the Christoffel numbers, xjn are the zeros of the orthonormal polynomials for the weight W2, and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξnn1/3, where ξn → ∞ as n → ∞. Previous estimates could sum only over those xjn with ¦xjn¦ σx1n, some fixed 0 < σ < 1. |
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Keywords: | Quadrature sums Freud weights Markov-Bernstein inequalities |
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