Two related objects are commonly called Hermite functions. In harmonic analysis, the 
th
Hermite function is the product of the 
th Hermite polynomial
and a Gaussian function. A common orthonormal
normalization is
 |
for 
,
1, 2, 
,
where 
is the 
th
Hermite polynomial. The functions 
form an orthogonal basis
for the Hilbert space 
. Under the symmetric angular-frequency convention, they
are eigenfunctions of the Fourier
transform. They also give the stationary-state solutions of the Schrödinger
equation for the quantum harmonic oscillator. Rescaling the variable and multiplying
by a nonzero constant gives other frequently used normalizations.
In special-function theory, the name Hermite function is also used for the extension 
of the Hermite polynomials to
noninteger order 
.
