Hermite functions
E502189
Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Hermite polynomials | 2 |
| Hermite function | 1 |
| Hermite functions canonical | 1 |
| Hermite polynomial | 1 |
| Hermite–Gaussian modes | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5191842 — 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: Hermite functions Context triple: [Charles Hermite, knownFor, Hermite functions]
-
A.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
B.
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.
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C.
Wightman functions
Wightman functions are vacuum expectation values of time-ordered products of quantum fields that rigorously encode the correlation structure and axiomatic foundations of relativistic quantum field theory.
-
D.
Daubechies wavelets
Daubechies wavelets are a family of compactly supported orthogonal wavelets widely used in signal processing and image compression for their efficient time-frequency localization.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- 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: Hermite functions Target entity description: Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
-
A.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
B.
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.
-
C.
Wightman functions
Wightman functions are vacuum expectation values of time-ordered products of quantum fields that rigorously encode the correlation structure and axiomatic foundations of relativistic quantum field theory.
-
D.
Daubechies wavelets
Daubechies wavelets are a family of compactly supported orthogonal wavelets widely used in signal processing and image compression for their efficient time-frequency localization.
-
E.
Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
family of orthogonal functions
ⓘ
special functions ⓘ |
| builtFrom |
Gaussian weight
ⓘ
Hermite polynomials ⓘ |
| definedOn | real line ⓘ |
| eigenfunctionsOf |
Fourier transform operator up to phase
ⓘ
harmonic oscillator differential operator ⓘ quantum harmonic oscillator Hamiltonian ⓘ |
| form |
Hilbert space basis
ⓘ
complete orthonormal system in L2(R) ⓘ |
| haveProperty |
rapid decay at infinity
ⓘ
real-valuedness for real arguments ⓘ smoothness ⓘ square integrability ⓘ |
| namedAfter | Charles Hermite ⓘ |
| normalizedVersionOf | Hermite polynomials times Gaussian ⓘ |
| orthogonalWithRespectTo |
Gaussian weight
ⓘ
Lebesgue measure with Gaussian weight ⓘ |
| relatedTo |
Fourier transform
ⓘ
Gaussian functions ⓘ Hermite polynomials ⓘ Hermite–Gaussian modes ⓘ
linked to:
Hermite functions
Weber–Hermite differential equation ⓘ
linked to:
Hermite differential equation
harmonic oscillator operator ⓘ parabolic cylinder functions ⓘ |
| satisfy |
Hermite differential equation in weighted form
ⓘ
completeness relations ⓘ orthogonality relations ⓘ recurrence relations ⓘ |
| usedIn |
Fourier analysis
ⓘ
Gabor analysis ⓘ Schrödinger equation ⓘ approximation theory ⓘ basis function expansions ⓘ harmonic analysis ⓘ image processing ⓘ numerical analysis ⓘ optics ⓘ orthogonal expansions on the real line ⓘ pattern recognition ⓘ probability theory ⓘ quantum harmonic oscillator ⓘ quantum mechanics ⓘ signal expansion ⓘ signal processing ⓘ solution of differential equations ⓘ spectral decomposition ⓘ spectral methods ⓘ stochastic processes ⓘ time-frequency analysis ⓘ wave packet analysis ⓘ |
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: Hermite functions Description of subject: Hermite functions are a family of orthogonal functions built from Hermite polynomials and a Gaussian weight, widely used in quantum mechanics, signal processing, and approximation theory.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hermite functions
linked to: Hermite functions
linked to: Hermite functions
linked to: Hermite functions
linked to: Hermite functions