Yesterday I was thinking about the uniqueness of the heat equation in both the Cauchy setting and the Dirichlet boundary setting. While thinking about this, the mathematics came down to computing the convolution of the heat kernel with the initial condition which is periodic with fundamental domain \([0,L]\).

After while, I needed to compute the convolution of the heat kernel with a sin function. In probabilistic terms, I need to compute

$$ E[sin(X)] \; X \sim N(\mu, \sigma) $$

Brute forcing the expected value is very difficult. Then i realized that we can use the good old characteristic function if we represent \(sin(X)\) as \(\frac{e^{iX}-e^{-iX}}{2i}\)

If we convolve a sine signal with some kernel, how do we know the output is still a sine signal? Well if the kernel is symmetric (even), that means the Fourier transform will be even. Since sine has a spectrum consisting of two Dirac delta functions at \(+\omega\) and \(-\omega\), then the output of the convolution will be another sine function of a different magnitude.