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Christoffel–Darboux formula

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In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878). It states that

where fj(x) is the jth term of a set of orthogonal polynomials of squared norm hj and leading coefficient kj.

There is also a "confluent form" of this identity by taking limit:

Proof

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Let be a sequence of polynomials orthonormal with respect to a probability measure , and define(they are called the "Jacobi parameters"), then we have the three-term recurrence[1]

Proof: By definition, , so if , then is a linear combination of , and thus . So, to construct , it suffices to perform Gram-Schmidt process on using , which yields the desired recurrence.


Proof of Christoffel–Darboux formula:

Since both sides are unchanged by multiplying with a constant, we can scale each to .

Since is a degree polynomial, it is perpendicular to , and so . Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.

Specific cases

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Hermite polynomials:

Associated Legendre polynomials:

See also

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References

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  1. ^ Świderski, Grzegorz; Trojan, Bartosz (2021-08-01). "Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I". Constructive Approximation. 54 (1): 49–116. arXiv:1909.09107. doi:10.1007/s00365-020-09519-w. ISSN 1432-0940. S2CID 202677666.