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Christoffel-Darboux Identity


\begin{displaymath}
\sum_{k=0}^\infty {\phi_k(x)\phi_k(y)\over\gamma_k}={\phi_{m+1}(x)\phi_m(y)-\phi_m(x)\phi_{m+1}(y)\over a_m\gamma_m(x-y),}
\end{displaymath} (1)

where $\phi_k(x)$ are Orthogonal Polynomials with Weighting Function $W(x)$,
\begin{displaymath}
\gamma_m\equiv \int [\phi_m(x)]^2W(x)\,dx,
\end{displaymath} (2)

and
\begin{displaymath}
a_k\equiv{A_{k+1}\over A_k}
\end{displaymath} (3)

where $A_k$ is the Coefficient of $x^k$ in $\phi_k(x)$.


References

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, p. 322, 1956.




© 1996-9 Eric W. Weisstein
1999-05-26