- Source: Hermite transform
In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials
H
n
(
x
)
{\displaystyle H_{n}(x)}
as kernels of the transform.
The Hermite transform
H
{
F
(
x
)
}
≡
f
H
(
n
)
{\displaystyle H\{F(x)\}\equiv f_{H}(n)}
of a function
F
(
x
)
{\displaystyle F(x)}
is
H
{
F
(
x
)
}
≡
f
H
(
n
)
=
∫
−
∞
∞
e
−
x
2
H
n
(
x
)
F
(
x
)
d
x
{\displaystyle H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}
The inverse Hermite transform
H
−
1
{
f
H
(
n
)
}
{\displaystyle H^{-1}\{f_{H}(n)\}}
is given by
H
−
1
{
f
H
(
n
)
}
≡
F
(
x
)
=
∑
n
=
0
∞
1
π
2
n
n
!
f
H
(
n
)
H
n
(
x
)
{\displaystyle H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}
Some Hermite transform pairs
References
Sources
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz [in German]; Tricomi, Francesco G. (1955), Higher transcendental functions (PDF), vol. II, McGraw-Hill, ISBN 978-0-07-019546-2, archived from the original (PDF) on 2011-07-14, retrieved 2023-11-09