Combination of line profiles

Question

When combining two line shapes (for example a Gaussian and a Lorentzian), the effect of Both of them combined is the convolution of both (with a Gaussian and a Lorentizan,this is the Voigt function). I am wondering what is the reason that a convolution arises.

I saw the explanation here: https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Stellar_Atmospheres_(Tatum)/10%3A_Line_Profiles/10.04%3A_Combination_of_Profiles

but it is not clear to me why

"At a distance $ξ $ from the line centre, the contribution to the line profile is the height of the function $𝑓(ξ)$ weighted by the function $𝑔(𝑥−ξ)$".

A clear explanation would be helpful.

Answer 1

Suppose we sample a function $f(x)$, but our resolution possesses a finite width $w=\Delta=const$: For each point $x_0$ we do not measure $f(x_0)$, but we obtain the moving average of the function $f$ with width $w$.

Next, we generalise this idea, and allow the finite resolution to be described by a function $w(x)$. We can think of it as a weighting function. At each point the same logic as above applies. However, instead of cutting out a finite part around $x=x_0$, and giving each point in this interval the same weight, the weight now depends of the distance to $x_0$. Thus, what we obtain is $$ g(x_0) = \int f(x-x_0) w(x) dx $$ which is identical to the convolution. Note, if we use a rectangular function to describe the finite resolution, we obtain the initially described moving average, where each point within a certain distance to $x_0$ gets the same weight. However, if we like to a continuous function to describe the contribution to each point, the Gaussian is a "natural" choice, because it possesses the largest entropy.