From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/hermite-polynomials/index.md | 94 ++++++++++++++++++++++++ 1 file changed, 94 insertions(+) create mode 100644 source/know/concept/hermite-polynomials/index.md (limited to 'source/know/concept/hermite-polynomials') diff --git a/source/know/concept/hermite-polynomials/index.md b/source/know/concept/hermite-polynomials/index.md new file mode 100644 index 0000000..17e61df --- /dev/null +++ b/source/know/concept/hermite-polynomials/index.md @@ -0,0 +1,94 @@ +--- +title: "Hermite polynomials" +date: 2021-09-08 +categories: +- Mathematics +- Statistics +layout: "concept" +--- + +The **Hermite polynomials** are a set of functions +that appear in physics and statistics, +although slightly different definitions are used in those fields. + + +## Physicists' definition + +The **Hermite equation** is an eigenvalue problem for $n$, +and the Hermite polynomials $H_n(x)$ are its eigenfunctions $u(x)$, +subject to the boundary condition that $u$ grows at most polynomially, +in which case the eigenvalues $n$ are non-negative integers: + +$$\begin{aligned} + \boxed{ + u'' - 2 x u' + 2 n u = 0 + } +\end{aligned}$$ + +The $n$th-order Hermite polynomial $H_n(x)$ +is therefore as follows, according to physicists: + +$$\begin{aligned} + H_n(x) + &= (-1)^n \exp(x^2) \dvn{n}{}{x}\exp(- x^2) + \\ + &= \Big( 2 x - \dv{}{x}\Big)^n 1 +\end{aligned}$$ + +This form is known as a *Rodrigues' formula*. +The first handful of Hermite polynomials are: + +$$\begin{gathered} + H_0(x) = 1 + \qquad \quad + H_1(x) = 2 x + \qquad \quad + H_2(x) = 4 x^2 - 2 + \\ + H_3(x) = 8 x^3 - 12 x + \qquad \quad + H_4(x) = 16 x^4 - 48 x^2 + 12 +\end{gathered}$$ + +And then more $H_n$ can be computed quickly +using the following recurrence relation: + +$$\begin{aligned} + \boxed{ + H_{n + 1}(x) = 2 x H_n(x) - 2n H_{n-1}(x) + } +\end{aligned}$$ + +They (almost) form an *Appell sequence*, +meaning their derivatives are like so: + +$$\begin{aligned} + \boxed{ + \dvn{k}{}{x}H_n(x) + = 2^k \frac{n!}{(n - k)!} H_{n - k}(x) + } +\end{aligned}$$ + +Importantly, all $H_n$ are orthogonal with respect to the weight function $w(x) \equiv \exp(- x^2)$: + +$$\begin{aligned} + \boxed{ + \Inprod{H_n}{w H_m} + \equiv \int_{-\infty}^\infty H_n(x) \: H_m(x) \: w(x) \dd{x} + = \sqrt{\pi} 2^n n! \: \delta_{nm} + } +\end{aligned}$$ + +Where $\delta_{nm}$ is the Kronecker delta. +Finally, they form a basis in the [Hilbert space](/know/concept/hilbert-space/) +of all functions $f(x)$ for which $\Inprod{f}{w f}$ is finite. +This means that every such $f$ can be expanded in $H_n$: + +$$\begin{aligned} + \boxed{ + f(x) + = \sum_{n = 0}^\infty a_n H_n(x) + = \sum_{n = 0}^\infty \frac{\Inprod{H_n}{w f}}{\Inprod{H_n}{w H_n}} H_n(x) + } +\end{aligned}$$ + -- cgit v1.2.3