Authors:
Z. Sebestyén Gödöllő University of Agricultural Sciences Department of Mathematics Faculty of Agricultural Engineering Páter K. U. 1 2103 Gödöllő Hungary

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Э. Себестяян Gödöllő University of Agricultural Sciences Department of Mathematics Faculty of Agricultural Engineering Páter K. U. 1 2103 Gödöllő Hungary

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Abstract  

Letx1, …,xn 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 putx0=0. In this paper, the following theorem is proved: Ify0, …,yn andy1, …,yn−1 are arbitrary real numbers, then there exists a unique polynomialP2n−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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Analysis Mathematica
Language English
Size B5
Year of
Foundation
1975
Volumes
per Year
1
Issues
per Year
4
Founder Akadémiai Kiadó
Founder's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0133-3852 (Print)
ISSN 1588-273X (Online)