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Christoffel Number

One of the quantities $\lambda_i$ appearing in the Gauss-Jacobi Mechanical Quadrature. They satisfy

\lambda_1+\lambda_2+\ldots+\lambda_n=\int_a^b d\alpha(x)=\alpha(b)-\alpha(a)
\end{displaymath} (1)

and are given by
$\displaystyle \lambda_\nu$ $\textstyle =$ $\displaystyle \int_a^b \left[{p_n(x)\over p_n'(x_\nu)(x-x_\nu)}\right]^2\,d\alpha(x)$ (2)
$\displaystyle \lambda_\nu$ $\textstyle =$ $\displaystyle -{k_{n+1}\over k_n}{1\over p_{n+1}(x_\nu)p_n'(x_\nu)}$ (3)
  $\textstyle =$ $\displaystyle {k_n\over k_{n-1}} {1\over p_{n-1}(x_\nu)P_n'(x_\nu)}$ (4)
$\displaystyle (\lambda_\nu)^{-1}$ $\textstyle =$ $\displaystyle [p_0(x_\nu)]^2+\ldots+[p_n(x_\nu)]^2,$ (5)

where $k_n$ is the higher Coefficient of $p_n(x)$.


Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 47-48, 1975.

© 1996-9 Eric W. Weisstein