Christoffel–Darboux formula

E947533

mathematical formula result in orthogonal polynomial theory

The Christoffel–Darboux formula is a key result in the theory of orthogonal polynomials that provides an explicit expression for sums of products of such polynomials, with important applications in approximation theory and mathematical physics.

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All labels observed (2)

Label	Occurrences
Christoffel–Darboux formula canonical	1
Christoffel–Darboux kernel	1

Statements (47)

Predicate	Object
instanceOf	mathematical formula ⓘ result in orthogonal polynomial theory ⓘ
appliesTo	Chebyshev polynomials NERFINISHED ⓘ Hermite polynomials NERFINISHED ⓘ Jacobi polynomials NERFINISHED ⓘ Laguerre polynomials NERFINISHED ⓘ Legendre polynomials NERFINISHED ⓘ families of polynomials satisfying a three-term recurrence ⓘ orthogonal polynomials on the unit circle ⓘ orthogonal polynomials with respect to a measure ⓘ
context	Hilbert space of square-integrable functions ⓘ reproducing kernel Hilbert spaces ⓘ
describes	sum of products of orthogonal polynomials ⓘ
field	analysis ⓘ approximation theory ⓘ mathematical physics ⓘ mathematics ⓘ orthogonal polynomials ⓘ random matrix theory ⓘ spectral theory ⓘ
gives	closed form for kernel of orthogonal polynomials ⓘ expression for partial sums of orthogonal expansions ⓘ
hasVariant	continuous Christoffel–Darboux formula NERFINISHED ⓘ discrete Christoffel–Darboux formula ⓘ multivariate Christoffel–Darboux formula NERFINISHED ⓘ
involves	orthogonality measure ⓘ three-term recurrence coefficients of orthogonal polynomials ⓘ
namedAfter	Elwin Bruno Christoffel NERFINISHED ⓘ Gaston Darboux NERFINISHED ⓘ
relatedTo	Gaussian quadrature ⓘ orthogonal polynomial ensembles ⓘ random matrix kernels ⓘ reproducing kernel ⓘ spectral methods ⓘ three-term recurrence relation ⓘ
relates	orthogonal polynomials of consecutive degrees ⓘ reproducing kernels of polynomial subspaces ⓘ
usedFor	Gaussian quadrature error analysis ⓘ analysis of convergence of orthogonal series ⓘ analysis of interpolation processes ⓘ approximation of functions by orthogonal polynomials ⓘ asymptotic analysis of orthogonal polynomials ⓘ construction of Christoffel–Darboux kernels ⓘ derivation of universality limits in random matrix theory ⓘ spectral approximation of differential operators ⓘ study of eigenvalue distributions in random matrices ⓘ study of zeros of orthogonal polynomials ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Elwin Bruno Christoffel → notableWork → Christoffel–Darboux formula ⓘ

this entity surface form: Christoffel–Darboux kernel