Christoffel-Darboux formula

This is the original Christoffel-Darboux formula that includes only the polynomials of the first kind [47,48]. The formula (151) with mixed polynomials Pn and Qm has no meaning at u = u. However, this is not so in the case of Eq. (152), which is well defined in the limit u —> u. The indeterminate expression 0/0 is regularized by the THopital rule which gives ... 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

We wish to add now another important property of orthogonal poly-nomials. " From the recurrence formula (7.10) the Christoffel-Darboux relation can be easily deduced ... 

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