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  • Author or Editor: Z. Sebestyén x
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Abstract  

Letx 1, …,x n be givenn distinct positive nodal points which generate the polynomial

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\omega _n (x) = \prod\limits_{i = 1}^n {(x - x_i )} .$$ \end{document}
Letx*1, …,x*n−1 be the roots of the derivativeωn(x) and putx 0=0. In this paper, the following theorem is proved: Ify 0, …,y n andy1, …,yn−1 are arbitrary real numbers, then there exists a unique polynomialP 2n−1(x) of degree 2n−1 having the following interpolation properties:
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_{2n - 1} (x_j ) = y_j (j = 0,...,n),$$ \end{document}
,
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_{2n - 1}^\prime (x_j^* ) = y_j^\prime (j = 1,...,n - 1).$$ \end{document}
. This result gives the theoretical completion of the original Pál type interpolation process, since it ensures uniqueness without assuming any additional condition.

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Abstract  

Let

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P_n (x) = \frac{{( - 1)^n }}{{2^n n!}}\frac{{d^n }}{{dx^n }}\left[ {(1 - x^2 )^n } \right]$$ \end{document}
be thenth Legendre polynomial. Letx 1,x 2,…,x n andx*1,x*2,…,x*n−1 denote the roots ofP n(x) andP′ n(x), respectively. Putx 0=x*0=−1 andx*n=1. In this paper we prove the following theorem: Ify 0,y 1,…,y n andy′ 0,y′ 1, …,y′ n are two systems of arbitrary real numbers, then there exists a unique polynomialQ 2n+1(x) of degree at most 2n+1 satisfying the conditions
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$Q_{2n + 1} (x_k^* ) = y_k and Q_{2n + 1}^\prime (x_k ) = y_k^\prime (k = 0,...,n).$$ \end{document}
.

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Abstract  

Let −1<α≤0 and let

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$L_n^{(\alpha )} (x) = \frac{1}{{n!}}x^{ - \alpha } e^x \frac{{d^n }}{{dx^n }}(x^{\alpha + n} e^{ - x} )$$ \end{document}
be the generalizednth Laguerre polynomial,n=1,2,… Letx 1,x 2,…,x n andx*1,x*2,…,x*n−1 denote the roots ofL n (α) (x) andL n (α)′ (x) respectively and putx*0=0. In this paper we prove the following theorem: Ify 0,y 1,…,y n−1 andy 1 ,…,y n are two systems of arbitrary real numbers, then there exists a unique polynomialP(x) of degree 2n−1 satisfying the conditions
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} P\left( {x_k^* } \right) = y_k (k = 0,...,n - 1) \hfill \\ P'\left( {x_k } \right) = y_k^\prime (k = 1,...,n). \hfill \\ \end{gathered}$$ \end{document}
.

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