Application of the Szili method of interpolation on the roots of the Legendre polynomials |
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Authors: | Z F Sebestyén З Ф Щебещтяен |
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Institution: | 1. Department of Mathematics Faculty of Agricultural Engineering, G?d?ll? University of Agricultural Sciences, Páter K. U. 1, 2103, G?d?ll?, Hungary
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Abstract: | Let $$P_n (x) = \frac{{( - 1)^n }}{{2^n n!}}\frac{{d^n }}{{dx^n }}\left {(1 - x^2 )^n } \right]$$ 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 $$Q_{2n + 1} (x_k^* ) = y_k and Q_{2n + 1}^\prime (x_k ) = y_k^\prime (k = 0,...,n).$$ . |
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