The assignment I'm solving right now is to prove that the computation of a FWHM of a laser line can be expressed as a convolution with some kernel function. And the purpose is fast peak detection. Although in reality we're working with frequency, this is not necessary for the math and we can abstract the variable as "x". The fact that we're dealing with the spectrum of a laser beam is just as irrelevant - the task is to find the peak position $x_c$ of a gaussian. Calculating the derivative and setting it to zero works as well, not sure why it's not sufficient - it has probably something to do with noise, but anyways.. The idea of FWHM peak position detection is to take two points of equal magnitude (they don't have to be at half the maximum magnitude necessarily) and then to find the maximum as their average.
The hint was: use convolution properties!
I found this kernel: $$g(x) = \frac{\delta(x) + \delta(x-w)}{2}$$
So 2 delta functions with half the mass, where w is the expected FWHM width.
Convolving with a delta gives you the function at that point: $$\int_{-\infty}^{\infty}f(t)\delta(\tau-t)dt=f(\tau)$$ I am not sure if this counts as a convolution property mentioned in the Hint.
Convolution with a delta and a shifted delta should give you a sum of the two points, then. And if both have half the weight, then this is equivalent to an average of the two points. And the maximum of the convolution function should be right where the two points are equal. Recording the positions of the points and taking their average gives you the peak. What do you think? Thanks!