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Abstract

Let μ be a compactly suppported positive measure on the real line. A point x∊supp [μ] is said to be μ-regular, if, as n→∞,
ea
Otherwise it is a μ-irregular point. We show that for any such measure, the set of μ-irregular points in {μ′>0} (with a suitable definition of this set) has Hausdorff measure 0, for , any β>1.
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Abstract  

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L 2 norm.

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Abstract

Müntz–Legendre polynomials L n(Λ;x) associated with a sequence Λ={λ k} are obtained by orthogonalizing the system in L 2[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is
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where and J 0 is the Bessel function of order 0.
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