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Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the Čebyšev roots) not describing the derivative values at “exceptional” nodes {ηn}n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever −1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [−1, 1].

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In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L p-spaces.

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Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E n(f) and E n (2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E n (2) (f), and Lorentz and Zeller proved that the inverse inequality E n (2) (f) ≦ cE n(f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ2. In this paper we prove, for every α > 0 and function f ∈ Δ2, that

\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} $$\sup \{ n^\alpha E_n^{(2)} (f):n \in \mathbb{N}\} \leqq c(\alpha )\sup \{ n^\alpha E_n (f):n \in \mathbb{N}\} ,$$ \end{document}
where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases s≦ 1 and s ≧ 2.

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The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l 2-norm for some general Mercer kernel matrices are provided. As an example, we give the l 2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l 2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

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We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.

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The aim of this paper is to give weighted function spaces in which the sequence of Cesàro means of the Jacobi-Fourier series are uniformly convergent. Error estimate for the approximation will also be considered.

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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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We give a weighted Hermite-Fejr-type interpolatory method on the real line, which is a positive operator on “good” matrices. We give an example on “good” interpolatory matrix by weighted Fekete points. To prove the convergence theorem we need the generalization of “Rodrigues’ property”.

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Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ n (α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w (a,b) σ n (α,β)Cw (a,b) f, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.

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Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y iY:= {y i}i∈ℤ such that for x ∈ [y i, y i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each nN(Y) we construct a trigonometric polynomial P n of order ≦ n, changing its sign at the same points y iY as f, and

\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} $$\left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right),$$ \end{document}
where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.

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