Gauss-Hermite

Convolution and Integration

The Gauss-Hermite (GH) expansion is a standard parametrization for galaxy emission-line kinematics, introduced by van der Marel & Franx (1993) and Gerhard (1993). It represents a profile that departs from a pure Gaussian through higher-order shape corrections, with \(h_3\) controlling skewness and \(h_4\) controlling kurtosis. This document derives the closed-form pixel integral used in integrate_gaussHermite() and shows how the LSF convolution reduces to a simple rescaling of the shape parameters.

The Gauss-Hermite Distribution

Using the probabilists’ Hermite polynomials \(\text{He}_m\), the normalised GH profile centred at \(\mu\) with width \(\sigma\) is

\[ \mathcal{L}(x) = \frac{e^{-y^2/2}}{\sigma\sqrt{2\pi}} \left[ 1 + \sum_{m=3}^{M} \frac{h_m}{\sqrt{m!}}\,\text{He}_m(y) \right], \quad y = \frac{x - \mu}{\sigma}. \]

The first few probabilists’ Hermite polynomials are

\[ \text{He}_2(y) = y^2 - 1,\quad \text{He}_3(y) = y^3 - 3y,\quad \text{He}_4(y) = y^4 - 6y^2 + 3. \]

The prefactor \(1/\sqrt{m!}\) ensures orthonormality with respect to the standard normal weight \(e^{-y^2/2}\), which guarantees that the profile integrates to 1 for any values of \(h_m\). The sum starts at \(m = 3\) because \(\text{He}_0 = 1\), \(\text{He}_1\), and \(\text{He}_2\) are absorbed into the normalisation, mean, and variance of the base Gaussian.

In unite the expansion is truncated at \(M = 4\), with free parameters \(h_3\) and \(h_4\).

Convolution with a Gaussian LSF

The instrumental line spread function (LSF) is modelled as a Gaussian with width \(\sigma_\text{lsf}\). Convolving \(\mathcal{L}\) with this kernel:

\[ (\mathcal{L} * G_\text{lsf})(x) = \int_{-\infty}^{\infty} \mathcal{L}(x')\,G_\text{lsf}(x - x')\,dx', \quad G_\text{lsf}(x) = \frac{e^{-x^2/(2\sigma_\text{lsf}^2)}}{\sigma_\text{lsf}\sqrt{2\pi}}. \]

Step 1 — Gaussian term. The zeroth-order term convolvs trivially:

\[ (G_\sigma * G_\text{lsf})(x) = G_{\sigma_\text{new}}(x), \quad \sigma_\text{new} = \sqrt{\sigma^2 + \sigma_\text{lsf}^2}. \]

Step 2 — Derivative identity. The probabilists’ Hermite polynomials satisfy the Rodrigues formula \(\text{He}_m(t)\,e^{-t^2/2} = (-1)^m\,d^m e^{-t^2/2}/dt^m\), from which:

\[ \text{He}_m\!\left(\frac{x}{\sigma}\right) G_\sigma(x) = (-\sigma)^m \frac{d^m G_\sigma}{dx^m}(x). \]

Step 3 — Integration by parts. Applying the identity to each Hermite correction and integrating by parts \(m\) times (boundary terms vanish since Gaussians decay at \(\pm\infty\)):

\[ \int G_\sigma(x')\,\text{He}_m\!\left(\frac{x'}{\sigma}\right) G_\text{lsf}(x-x')\,dx' = \sigma^m \frac{d^m G_{\sigma_\text{new}}}{dx^m}(x). \]

Step 4 — Apply the identity in reverse. Converting the derivative back to a Hermite polynomial using the Rodrigues formula:

\[ \sigma^m \frac{d^m G_{\sigma_\text{new}}}{dx^m}(x) = \left(\frac{\sigma}{\sigma_\text{new}}\right)^m \text{He}_m(y_\text{new})\,G_{\sigma_\text{new}}(x), \quad y_\text{new} = \frac{x - \mu}{\sigma_\text{new}}. \]

Result. The convolved distribution is again a GH profile with the same \(\mu\) but wider width \(\sigma_\text{new}\) and rescaled shape parameters:

\[ (\mathcal{L} * G_\text{lsf})(x) = \frac{e^{-y_\text{new}^2/2}}{\sigma_\text{new}\sqrt{2\pi}} \left[ 1 + \sum_{m=3}^{M} \frac{h_m'}{\sqrt{m!}}\, \text{He}_m(y_\text{new}) \right], \]
\[ h_m' = h_m \left(\frac{\sigma}{\sigma_\text{new}}\right)^m = h_m\,r^m, \quad r = \frac{\sigma}{\sigma_\text{new}} < 1. \]

The ratio \(r^m < 1\) suppresses higher-order moments: convolution with the LSF washes out the non-Gaussian character of the line, with stronger suppression for higher-order terms.

CDF and Pixel Integration

Exact pixel integration requires the cumulative distribution function (CDF) \(F(x) = \int_{-\infty}^x (\mathcal{L} * G_\text{lsf})(x')\,dx'\).

The Gaussian base term integrates directly to the standard normal CDF \(\Phi(y_\text{new})\). For each Hermite correction, the identity \(\int g(y)\,\text{He}_m(y)\,dy = -g(y)\,\text{He}_{m-1}(y) + C\) (the indefinite integral of a Gaussian-weighted Hermite polynomial reduces to the next-lower-order polynomial times the Gaussian envelope) yields:

\[ \int_{-\infty}^x G_{\sigma_\text{new}}(x')\,\text{He}_m(y')\,dx' = -\frac{g(y_\text{new})}{\sqrt{2\pi}}\,\text{He}_{m-1}(y_\text{new}), \quad g(y) = e^{-y^2/2}. \]

Collecting all terms:

\[ \boxed{ F(x) = \Phi(y_\text{new}) - \frac{g(y_\text{new})}{\sqrt{2\pi}} \sum_{m=3}^{M} \frac{h_m'}{\sqrt{m!}}\,\text{He}_{m-1}(y_\text{new}) } \]

Each correction term in the CDF is a Gaussian envelope \(g(y)\) multiplied by a polynomial of one order lower than the corresponding PDF correction. The pixel integral over the bin \([\lambda_l, \lambda_h]\) is then simply \(F(\lambda_h) - F(\lambda_l)\).

Connection to the Code

In integrate_gaussHermite(), the half-variance coordinate \(t = y/\sqrt{2}\) is used throughout so that erf can be called directly. Translating the CDF formula for \(m = 3, 4\):

\[ c_3 = \frac{h_3 r^3}{\sqrt{3!}} = \frac{h_3 r^3}{\sqrt{6}}, \quad c_4 = \frac{h_4 r^4}{\sqrt{4!}} = \frac{h_4 r^4}{\sqrt{24}}, \]

which correspond to c3 = h3 * r3 / _SQRT6 and c4 = h4 * r3 * r / _SQRT24 in the implementation. The antiderivative at coordinate \(y = t\sqrt{2}\) is:

\[ \text{\_integrandGH}(t; c_3, c_4) = g(y)\bigl[c_3\,\text{He}_2(y) + c_4\,\text{He}_3(y)\bigr] = e^{-y^2/2}\bigl[c_3(y^2-1) + c_4\,y(y^2-3)\bigr], \]

and the final pixel integral is:

\[ \frac{\text{erf}(t_h) - \text{erf}(t_l)}{2} - \frac{1}{\sqrt{2\pi}} \bigl[\text{\_integrandGH}(t_h) - \text{\_integrandGH}(t_l)\bigr]. \]