The orthogonal and the nodal polynomials

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.