In mathematics, the Hermite transform is an

integral transform In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily charac ...

named after the mathematician

Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite p ...

that uses

Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...

H_n(x)

as kernels of the transform. The Hermite transform

H\ \equiv f_H (n)

of a function

F(x)

H\ \equiv f_H(n) = \int_^\infty e^ \ H_n(x)\  F(x) \ dx

The inverse Hermite transform

H^\

is given by

H^\ \equiv F(x) = \sum_^\infty \frac f_H(n) H_n(x)

Some Hermite transform pairs

References

Sources

*{{citation , last1=Erdélyi , first1=Arthur , author-link1=Arthur Erdélyi , last2=Magnus , first2=Wilhelm , author-link2=Wilhelm Magnus , last3=Oberhettinger , first3=Fritz , author-link3=:de:Fritz Oberhettinger , last4=Tricomi , first4=Francesco G. , author-link4=Francesco Tricomi , title=Higher transcendental functions , volume=II , publisher=McGraw-Hill , year=1955 , url=http://apps.nrbook.com/bateman/Vol2.pdf , isbn=978-0-07-019546-2 , access-date=2023-11-09 , archive-date=2011-07-14 , archive-url=https://web.archive.org/web/20110714210423/http://apps.nrbook.com/bateman/Vol2.pdf , url-status=dead Integral transforms Mathematical physics