The orthogonal and the nodal polynomials

Yu. I. Kuznetsov

Institute of Computational Mathematics and Mathematical Geophysics (Computing Center), Siberian Branch of the Russian Academy of Sciences

Abstract: The polynomials $P_k(x)$ of the degree $k$ that are orthogonal on a finite set of the points $x_i$, $i=1(1)n$, with weights $c_i>0$, are considered. It is shown that the polynomial $P_k(x)$ is a linear functional of the nodal polynomials of the same degree, expressed by $x_i$, $c_i$. The vector that defines this functional is positive and normalized. Such properties of the functional describe it as average, or the center of mass, of the nodal polynomials distributed with the corresponding density.

Key words: polynomial, orthogonal, nodes, finite, set, liner, functional, average, density.

UDC: 517.518.36

Received: 15.06.2005
Revised: 08.09.2005