Hermite transform

In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials $H_{n}(x)$ as kernels of the transform.

The Hermite transform $H\{F(x)\}\equiv f_{H}(n)$ of a function $F(x)$ is $$H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx$$

The inverse Hermite transform $H^{-1}\{f_{H}(n)\}$ is given by $$H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)$$

Computational complexity

Direct evaluation of a discrete Hermite transform with $N$ coefficients and $O(N)$ sample points requires $O(N^{2})$ arithmetic operations. Leibon, Rockmore, Park, Taintor, and Chirikjian developed fast algorithms for the forward and inverse transforms requiring $O(N\log ^{2}N)$ operations, exploiting the three-term recurrence satisfied by the Hermite polynomials.

Jain, Iyer, Somma, Bao, and Jordan developed a quantum algorithm that implements an approximate discrete Hermite transform using a number of quantum gates polynomial in $\log N$ and $\log(1/\varepsilon )$, where $N$ is the number of Hermite modes and $\varepsilon$ is the approximation error. The algorithm uses fast-forwarding of the quantum harmonic oscillator. It transforms information encoded in the amplitudes of a quantum state, rather than producing an explicit classical list of transform coefficients; the gate bound does not include preparing an arbitrary input state from classical data.

Some Hermite transform pairs

$F(x)\,$	$f_{H}(n)\,$
$x^{m}$	${\begin{cases}{\frac {m!{\sqrt {\pi }}}{2^{m-n}\left({\frac {m-n}{2}}\right)!}},&(m-n){\text{ even and}}\geq 0\\0,&{\text{otherwise}}\end{cases}}$
$e^{ax}\,$	${\sqrt {\pi }}a^{n}e^{a^{2}/4}\,$
$e^{2xt-t^{2}},\ |t|<{\frac {1}{2}}\,$	${\sqrt {\pi }}(2t)^{n}$
$H_{m}(x)\,$	${\sqrt {\pi }}2^{n}n!\delta _{nm}\,$
$x^{2}H_{m}(x)\,$	$2^{n}n!{\sqrt {\pi }}{\begin{cases}1,&n=m+2\\\left(n+{\frac {1}{2}}\right),&n=m\\(n+1)(n+2),&n=m-2\\0,&{\text{otherwise}}\end{cases}}$
$e^{-x^{2}}H_{m}(x)\,$	$\left(-1\right)^{p-m}2^{p-1/2}\Gamma (p+1/2),\ m+n=2p,\ p\in \mathbb {Z}$
$H_{m}^{2}(x)\,$	${\begin{cases}2^{m+n/2}{\sqrt {\pi }}{\binom {m}{n/2}}{\frac {m!n!}{(n/2)!}},&n{\text{ even and}}\leq 2m\\0,&{\text{otherwise}}\end{cases}}$
$H_{m}(x)H_{p}(x)\,$	${\begin{cases}{\frac {2^{k}{\sqrt {\pi }}m!n!p!}{(k-m)!(k-n)!(k-p)!}},&n+m+p=2k,\ k\in \mathbb {Z$ ;\ |m-p|\leq n\leq m+p\\0,&{\text{otherwise}}\end{cases}}\,}
$H_{n+p+q}(x)H_{p}(x)H_{q}(x)\,$	${\sqrt {\pi }}2^{n+p+q}(n+p+q)!\,$
${\frac {d^{m}}{dx^{m}}}F(x)\,$	$f_{H}(n+m)\,$
$x{\frac {d^{m}}{dx^{m}}}F(x)\,$	$nf_{H}(n+m-1)+{\frac {1}{2}}f_{H}(n+m+1)\,$
$e^{x^{2}}{\frac {d}{dx}}\left[e^{-x^{2}}{\frac {d}{dx}}F(x)\right]\,$	$-2nf_{H}(n)\,$
$F(x-x_{0})$	${\sqrt {\pi }}\sum _{k=0}^{\infty }{\frac {(-x_{0})^{k}}{k!}}f_{H}(n+k)$
$F(x)*G(x)\,$	${\sqrt {\pi }}(-1)^{n}\left[2^{2n+1}\Gamma \left(n+{\frac {3}{2}}\right)\right]^{-1}f_{H}(n)g_{H}(n)\,$
$e^{z^{2}}\sin(xz),\ |z|<{\frac {1}{2}}\ \,$	${\begin{cases}{\sqrt {\pi }}(-1)^{\lfloor {\frac {n}{2}}\rfloor }(2z)^{n},&n\,\mathrm {odd} \\0,&n\,\mathrm {even} \end{cases}}\,$
$(1-z^{2})^{-1/2}\exp \left[{\frac {2xyz-(x^{2}+y^{2})z^{2}}{(1-z^{2})}}\right]\,$	${\sqrt {\pi }}z^{n}H_{n}(y)$
${\frac {H_{m}(y)H_{m+1}(x)-H_{m}(x)H_{m+1}(y)}{2^{m+1}m!(x-y)}}$	${\begin{cases}{\sqrt {\pi }}H_{n}(y)&n\leq m\\0&n>m\end{cases}}$

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 {{citation}}: ISBN / Date incompatibility (help)