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 $H {F (x)} \equiv f_{H} (n) = \int_{- \infty}^{\infty} e^{- x^{2}} H_{n} (x) F (x) d x$

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

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.^[1]

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 $\log (1 / ε)$ , where $N$ is the number of Hermite modes and $ε$ 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.^[2]

Some Hermite transform pairs

$F (x)$	$f_{H} (n)$
$x^{m}$	${\begin{matrix} \frac{m! \sqrt{π}}{2^{m - n} (\frac{m - n}{2})!}, & (m - n) even and \geq 0 \\ 0, & otherwise \end{matrix}$ ^[3]
$e^{a x}$	$\sqrt{π} a^{n} e^{a^{2} / 4}$
$e^{2 x t - t^{2}}, \| t \| < \frac{1}{2}$	$\sqrt{π} (2 t)^{n}$
$H_{m} (x)$	$\sqrt{π} 2^{n} n! δ_{n m}$
$x^{2} H_{m} (x)$	$2^{n} n! \sqrt{π} {\begin{matrix} 1, & n = m + 2 \\ (n + \frac{1}{2}), & n = m \\ (n + 1) (n + 2), & n = m - 2 \\ 0, & otherwise \end{matrix}$
$e^{- x^{2}} H_{m} (x)$	${(- 1)}^{p - m} 2^{p - 1 / 2} Γ (p + 1 / 2), m + n = 2 p, p \in ℤ$
$H_{m}^{2} (x)$	${\begin{matrix} 2^{m + n / 2} \sqrt{π} (\binom{m}{n / 2}) \frac{m! n!}{(n / 2)!}, & n even and \leq 2 m \\ 0, & otherwise \end{matrix}$ ^[4]
$H_{m} (x) H_{p} (x)$	${\begin{matrix} \frac{2^{k} \sqrt{π} m! n! p!}{(k - m)! (k - n)! (k - p)!}, & n + m + p = 2 k, k \in ℤ; \| m - p \| \leq n \leq m + p \\ 0, & otherwise \end{matrix}$ ^[5]
$H_{n + p + q} (x) H_{p} (x) H_{q} (x)$	$\sqrt{π} 2^{n + p + q} (n + p + q)!$
$\frac{d^{m}}{d x^{m}} F (x)$	$f_{H} (n + m)$
$x \frac{d^{m}}{d x^{m}} F (x)$	$n f_{H} (n + m - 1) + \frac{1}{2} f_{H} (n + m + 1)$
$e^{x^{2}} \frac{d}{d x} [e^{- x^{2}} \frac{d}{d x} F (x)]$	$- 2 n f_{H} (n)$
$F (x - x_{0})$	$\sqrt{π} \sum_{k = 0}^{\infty} \frac{(- x_{0})^{k}}{k!} f_{H} (n + k)$
$F (x) * G (x)$	$\sqrt{π} (- 1)^{n} {[2^{2 n + 1} Γ (n + \frac{3}{2})]}^{- 1} f_{H} (n) g_{H} (n)$ ^[6]
$e^{z^{2}} \sin (x z), \| z \| < \frac{1}{2}$	${\begin{matrix} \sqrt{π} (- 1)^{⌊ \frac{n}{2} ⌋} (2 z)^{n}, & n o d d \\ 0, & n e v e n \end{matrix}$
$(1 - z^{2})^{- 1 / 2} \exp [\frac{2 x y z - (x^{2} + y^{2}) z^{2}}{(1 - z^{2})}]$	$\sqrt{π} z^{n} H_{n} (y)$ ^[7]^[8]
$\frac{H_{m} (y) H_{m + 1} (x) - H_{m} (x) H_{m + 1} (y)}{2^{m + 1} m! (x - y)}$	${\begin{matrix} \sqrt{π} H_{n} (y) & n \leq m \\ 0 & n > m \end{matrix}$

References

↑ Leibon, Gregory; Rockmore, Daniel N.; Park, Wooram; Taintor, Robert; Chirikjian, Gregory S. (2008). "A fast Hermite transform". Theoretical Computer Science. 409 (2): 211–228. doi:10.1016/j.tcs.2008.09.010. PMC 2630232.
↑ Jain, Siddhartha; Iyer, Vishnu; Somma, Rolando D.; Bao, Ning; Jordan, Stephen P. (2026). "Efficient Quantum Hermite Transform". Proceedings of the 58th Annual ACM Symposium on Theory of Computing. Association for Computing Machinery. pp. 541–552. arXiv:2510.04929. doi:10.1145/3798129.3800772.
↑ McCully, Joseph Courtney; Churchill, Ruel Vance (1953), Hermite and Laguerre integral transforms : preliminary report
↑ Feldheim, Ervin (1938). "Quelques nouvelles relations pour les polynomes d'Hermite". Journal of the London Mathematical Society (in French). s1-13: 22–29. doi:10.1112/jlms/s1-13.1.22.
↑ Bailey, W. N. (1939). "On Hermite polynomials and associated Legendre functions". Journal of the London Mathematical Society. s1-14 (4): 281–286. doi:10.1112/jlms/s1-14.4.281.
↑ Glaeske, Hans-Jürgen (1983). "On a convolution structure of a generalized Hermite transformation" (PDF). Serdica Bulgariacae Mathematicae Publicationes. 9 (2): 223–229.
↑ Erdélyi et al. 1955, p. 194, 10.13 (22).
↑ Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions], Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).

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)

[Leibon2008-1] Leibon, Gregory; Rockmore, Daniel N.; Park, Wooram; Taintor, Robert; Chirikjian, Gregory S. (2008). "A fast Hermite transform". Theoretical Computer Science. 409 (2): 211–228. doi:10.1016/j.tcs.2008.09.010. PMC 2630232.

[Jain2026-2] Jain, Siddhartha; Iyer, Vishnu; Somma, Rolando D.; Bao, Ning; Jordan, Stephen P. (2026). "Efficient Quantum Hermite Transform". Proceedings of the 58th Annual ACM Symposium on Theory of Computing. Association for Computing Machinery. pp. 541–552. arXiv:2510.04929. doi:10.1145/3798129.3800772.

[3] McCully, Joseph Courtney; Churchill, Ruel Vance (1953), Hermite and Laguerre integral transforms : preliminary report

[4] Feldheim, Ervin (1938). "Quelques nouvelles relations pour les polynomes d'Hermite". Journal of the London Mathematical Society (in French). s1-13: 22–29. doi:10.1112/jlms/s1-13.1.22.

[5] Bailey, W. N. (1939). "On Hermite polynomials and associated Legendre functions". Journal of the London Mathematical Society. s1-14 (4): 281–286. doi:10.1112/jlms/s1-14.4.281.

[6] Glaeske, Hans-Jürgen (1983). "On a convolution structure of a generalized Hermite transformation" (PDF). Serdica Bulgariacae Mathematicae Publicationes. 9 (2): 223–229.

[7] Erdélyi et al. 1955, p. 194, 10.13 (22).

[8] Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung" [On the development of a function of arbitrarily many variables according to higher-order Laplace functions], Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj. See p. 174, eq. (18) and p. 173, eq. (13).

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