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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}.
  1. ^ http://www.hep.caltech.edu/~fcp/math/differentialEquations/linDiffEq.pdf

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