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# Christoffel–Darboux formula: Wikis

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# Encyclopedia

In mathematics, the ChristoffelDarboux theorem converts a linear operator on a linear subspace into a symmetric kernel. From the projection operator,

$J_n(x,y) = \sum_{j=0}^n \frac{f_j(x) f_j(y)}{h_j}$

where fj(x) is the $j^{\,th}$ term of a set of orthogonal polynomials, results the Christoffel–Darboux Theorem[1]:

$J_n (x,y) = \frac{k_n}{h_n k_{n+1}} \frac{f_n(y) f_{n+1}(x) - f_{n+1}(y) f_n(x)}{x - y}.$