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The Hilbert Space Asymptotics of a Class of Orthonormal Polynomials on a Bounded Interval

  • S. N. M. Ruijsenaars
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Part of the Developments in Mathematics book series (DEVM, volume 13)

Abstract

We study the asymptotics of orthonormal polynomials {p n (cos x)} n=0 , associated with a certain class of weight functions \(w\left( \mathfrak{X} \right) = 1/c\left( \mathfrak{X} \right)c\left( { - \mathfrak{X}} \right)\) on [0, π]. Our principal result is that the norm of the difference of pn(cos x) and \(D_n \left( \mathfrak{X} \right) \equiv c\left( \mathfrak{X} \right)e^{in\mathfrak{X}} + c\left( { - \mathfrak{X}} \right)e^{ - in\mathfrak{X}} \) in L2([0, π], (2π)−1ω(x)dx) vanishes exponentially as n → ∞. The decay rate is determined by analyticity properties of the c-function.

Keywords

Weight Function Exponential Decay Orthogonal Polynomial Schwarz Inequality Jacobi Polynomial 
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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • S. N. M. Ruijsenaars
    • 1
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

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