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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Observed surface forms (3)

Surface form Occurrences
Hermite polynomials 2
Hermite function 1
Hermite polynomial 1

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 ⓘ

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 ⓘ
this entity surface form: Hermite polynomials