Hermite functions

E502189

family of orthogonal functions special functions

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.

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All labels observed (4)

Label	Occurrences
Hermite polynomials	2
Hermite function	1
Hermite functions canonical	1
Hermite polynomial	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.
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ
normalizedVersionOf	Hermite polynomials times Gaussian ⓘ
orthogonalWithRespectTo	Gaussian weight ⓘ Lebesgue measure with Gaussian weight ⓘ
relatedTo	Fourier transform NERFINISHED ⓘ Gaussian functions ⓘ Hermite polynomials NERFINISHED ⓘ Hermite–Gaussian modes NERFINISHED ⓘ Weber–Hermite differential equation NERFINISHED ⓘ harmonic oscillator operator ⓘ parabolic cylinder functions ⓘ
satisfy	Hermite differential equation in weighted form ⓘ completeness relations ⓘ orthogonality relations ⓘ recurrence relations ⓘ
usedIn	Fourier analysis NERFINISHED ⓘ Gabor analysis ⓘ Schrödinger equation NERFINISHED ⓘ 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 NERFINISHED ⓘ quantum mechanics NERFINISHED ⓘ 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

Referenced by (5)

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

Charles Hermite → knownFor → Hermite functions ⓘ

this entity surface form: Hermite polynomials

Charles Hermite → knownFor → Hermite functions ⓘ

Charles Hermite → hasConceptNamedAfter → Hermite functions ⓘ

this entity surface form: Hermite polynomial

Charles Hermite → hasConceptNamedAfter → Hermite functions ⓘ

this entity surface form: Hermite function

Edgeworth expansion → uses → Hermite functions ⓘ