Polynomial¶

Convolution with Gaussian LSF¶

Polynomial continuum forms (Linear, Polynomial, Chebyshev, Bernstein) are analytically convolved with the Gaussian instrumental LSF at evaluation time. This is possible because a polynomial convolved with a Gaussian is still a polynomial, with a coefficient transform that can be precomputed. This document derives that transform, which is implemented in _gaussian_convolve_poly().

Setup¶

Let the continuum model in some normalised coordinate \(x\) be a polynomial

\[ f(x) = \sum_{i=0}^{N} c_i\,x^i, \]

and let the LSF be a zero-mean Gaussian kernel with standard deviation \(\sigma\):

\[ G(x;\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-x^2/(2\sigma^2)}. \]

The convolved continuum is

\[ (f * G)(x) = \int_{-\infty}^{\infty} f(t)\,G(x-t;\sigma)\,dt. \]

Key Idea: Linearity and Moments¶

By linearity the convolution reduces to a sum over monomials:

\[ (f * G)(x) = \sum_{i=0}^{N} c_i \int_{-\infty}^{\infty} t^i\,G(x-t;\sigma)\,dt = \sum_{i=0}^N c_i\,M_i(x), \]

where \(M_i(x) = \int_{-\infty}^{\infty} t^i\,G(x-t;\sigma)\,dt\) is the \(i\)-th moment of the Gaussian centred at \(x\).

Substituting \(u = t - x\):

\[ M_i(x) = \int_{-\infty}^{\infty} (x+u)^i\,G(u;\sigma)\,du. \]

Expanding via the binomial theorem \((x+u)^i = \sum_{k=0}^i \binom{i}{k} x^{i-k} u^k\):

\[ M_i(x) = \sum_{k=0}^{i} \binom{i}{k} x^{i-k} \underbrace{\int_{-\infty}^{\infty} u^k\,G(u;\sigma)\,du}_{\mu_k}, \]

where \(\mu_k\) are the moments of a zero-mean Gaussian.

Gaussian Moments¶

All odd moments vanish by symmetry. The even moments are:

\[\begin{split} \mu_k = \begin{cases} 0 & k\text{ odd}, \\ (k-1)!!\;\sigma^k & k\text{ even}, \end{cases} \end{split}\]

where \((k-1)!! = 1 \cdot 3 \cdot 5 \cdots (k-1)\) is the double factorial (with the convention \((-1)!! = 1\), so \(\mu_0 = 1\)). For example: \(\mu_0 = 1\), \(\mu_2 = \sigma^2\), \(\mu_4 = 3\sigma^4\), \(\mu_6 = 15\sigma^6\).

The Convolved Polynomial¶

Substituting back and collecting terms by power of \(x\), the convolution is still a polynomial of the same degree \(N\):

\[ (f * G)(x) = \sum_{j=0}^{N} c_j^{\text{new}}\,x^j, \]

with new coefficients

\[\begin{split} \boxed{ c_j^{\text{new}} = \sum_{\substack{k=0\\k\text{ even}}}^{N-j} c_{j+k}\,\binom{j+k}{k}\,(k-1)!!\;\sigma^k. } \end{split}\]

Each output coefficient \(c_j^\text{new}\) receives contributions from all higher-order input coefficients \(c_{j+k}\) (for even \(k > 0\)): the LSF smears power from sharper features into smoother ones. The \(k = 0\) term is just \(c_j\) itself (LSF has no effect on a constant), and odd moments contribute nothing because \(G\) is symmetric.

Using FWHM Instead of \(\sigma\)¶

The Gaussian LSF is parametrised by its FWHM in unite:

\[ \sigma = \frac{\text{FWHM}}{2\sqrt{2\ln 2}}, \]

so the coefficient transform becomes

\[\begin{split} c_j^{\text{new}} = \sum_{\substack{k=0\\k\text{ even}}}^{N-j} c_{j+k}\,\binom{j+k}{k}\,(k-1)!! \left(\frac{\text{FWHM}}{2\sqrt{2\ln 2}}\right)^k. \end{split}\]

For basis-transformed forms (Chebyshev, Bernstein), the FWHM is first rescaled into the normalised coordinate domain before applying this transform, since the polynomial lives in \([-1,1]\) or \([0,1]\) rather than wavelength space.

Properties and Connection to the Code¶

The transform has three important properties:

Polynomial closure: \((f * G)\) is a polynomial of the same degree as \(f\) — no new basis functions are needed.
Only even moments contribute: odd-\(k\) terms vanish, so the sum steps by 2. This halves the number of non-zero terms for high-degree polynomials.
FWHM-controlled: the entire effect of the LSF enters through a single scalar \(\sigma\), computed from lsf_fwhm at each wavelength point.

In _gaussian_convolve_poly(), the coefficients are stored in NumPy descending-order convention (index 0 = highest power), so the index mapping is \(c_j \leftrightarrow \text{coeffs}[N-j]\). The even-moment double factorial is computed iteratively as \(\mu_{2j} = \mu_{2(j-1)} \cdot (2j-1)\sigma^2\), avoiding explicit factorial arithmetic. The result is a JAX array of convolved coefficients that can be evaluated with the same polynomial kernel used for the unconvolved form.