Christoffel–Darboux formula
E947533
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Christoffel–Darboux formula canonical | 3 |
| Christoffel–Darboux kernel | 1 |
| continuous Christoffel–Darboux formula | 1 |
| multivariate Christoffel–Darboux formula | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11812503 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Christoffel–Darboux formula Context triple: [Elwin Bruno Christoffel, notableWork, Christoffel–Darboux formula]
-
A.
Rodrigues formula
Rodrigues formula is a classical representation that expresses certain families of orthogonal polynomials, such as Jacobi polynomials, in terms of derivatives of weight functions.
-
B.
Gegenbauer polynomials
Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
-
C.
Jack polynomials
Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
-
D.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
E.
Cauchy–Binet formula
The Cauchy–Binet formula is a fundamental result in linear algebra that expresses the determinant of a product of two rectangular matrices as a sum of products of determinants of their square submatrices.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Christoffel–Darboux formula Target entity description: 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.
-
A.
Rodrigues formula
Rodrigues formula is a classical representation that expresses certain families of orthogonal polynomials, such as Jacobi polynomials, in terms of derivatives of weight functions.
-
B.
Gegenbauer polynomials
Gegenbauer polynomials are a family of orthogonal polynomials on the interval [-1, 1] that generalize Legendre polynomials and play a key role in harmonic analysis and solutions of differential equations with spherical symmetry.
-
C.
Jack polynomials
Jack polynomials are a family of symmetric polynomials depending on a continuous parameter that generalize several classical symmetric functions and play a key role in algebraic combinatorics, representation theory, and mathematical physics.
-
D.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
E.
Cauchy–Binet formula
The Cauchy–Binet formula is a fundamental result in linear algebra that expresses the determinant of a product of two rectangular matrices as a sum of products of determinants of their square submatrices.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in orthogonal polynomial theory ⓘ |
| appliesTo |
Chebyshev polynomials
ⓘ
linked to:
Chebyshev polynomials of the first kind
Hermite polynomials ⓘ Jacobi polynomials ⓘ Laguerre polynomials ⓘ Legendre polynomials ⓘ 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
ⓘ
linked to:
Christoffel–Darboux formula
discrete Christoffel–Darboux formula ⓘ multivariate Christoffel–Darboux formula ⓘ
linked to:
Christoffel–Darboux formula
|
| involves |
orthogonality measure
ⓘ
three-term recurrence coefficients of orthogonal polynomials ⓘ |
| namedAfter |
Elwin Bruno Christoffel
ⓘ
Gaston Darboux ⓘ |
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Christoffel–Darboux formula Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Christoffel–Darboux formula
subject linked to:
Christoffel
linked to: Christoffel–Darboux formula
linked to: Christoffel–Darboux formula