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The authors establish necessary and sufficient conditions for the weightedL p convergence at given rates of Hermite interpolation of higher order based on Jacobi zeros plus the endpoints ±1. Theorems on simultaneous approximation are also proved.

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The generalized Christoffel function λ p,q,n(;x) (0<p<∞, 0≦q<∞) with respect to a measure on R is defined by
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The novelty of our definition is that it contains the factor |tx|q, which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.

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For a measure μ on the complex plane μ-regular points play an important role in various polynomial inequalities. In the present work it is shown that every point in the set {μ′>0} (actually of a larger set where μ is strong) with the exception of a set of zero logarithmic capacity is a μ-regular point. Here “set of zero logarithmic capacity” cannot be replaced by “β-logarithmic Hausdorff measure 0” with β=1 (it can be replaced by “β-logarithmic measure 0” with any β>1). On the other hand, for arbitrary μ the set of μ-regular points can be quite small, but never empty.

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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→∞,
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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  

The paper is concerned with bounds for integrals of the type
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int\limits_{U_k (x)} {\left| {p_n^{(\alpha , \beta )} (t)} \right|^p w^{(a, b)} } (t) dt, p \geqq 0$$ \end{document}
, involving Jacobi polynomials p n (α,β) and Jacobi weights w (a,b) depending on α,β, a, b > −1, where the subsets U k(x) ⊂ [−1, 1] located around x and are given by
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$U_k (x) = \left[ {x - \tfrac{{\phi _k (x)}} {k}, x + \tfrac{{\phi _k (x)}} {k}} \right] \cap [ - 1, 1]$$ \end{document}
with
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\phi _k (x) = \sqrt {1 - x^2 } + \tfrac{1} {k}$$ \end{document}
. The functions to be integrated will also be of the type
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left| {\tfrac{{p_n^{(\alpha , \beta )} (t)}} {{x - t}}} \right|$$ \end{document}
on the domain [−1,1] t/ U k(x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in [3].
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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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We characterize the supports of the measures having quadrature formulae with similar exactness as Gauss’ theorem. Indeed we obtain the supports of the measures from which an m-point quadrature formula can be obtained such that it exactly integrates functions in the space ℙmk,mk[
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\bar z$$ \end{document}
, z]. We also give a method for obtaining the nodes and the quadrature coefficients in all the cases and, as a consequence, we solve the same problem in the space of trigonometric polynomials.
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Abstract  

We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of Al-Salam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.

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