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Hermite transform

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Source updated September 16, 2026

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 $H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)$

01Computational 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.

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