Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Question

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form $\displaystyle \frac{1}{\sqrt{2\pi}}a_0+\frac{1}{\sqrt{\pi}}\sum_{k=1}^\infty \left(a_k \cos kx+b_k \sin kx\right)$? Is it because of the Expansion theorem?

Answer 1

I don't fully understand the question: if you agree that $\left\{\frac{1}{\sqrt{2\pi}}, \frac{1}{\sqrt{\pi}}\cos kx, \frac{1}{\sqrt{\pi}}\sin kx\right\}$ are a basis for the function space of $2\pi$-periodic functions, it follows immediately that any function in this space can be expressed as a linear combination of these basis functions (and the $a_i$ and $b_i$ are these coefficients).... that's just what it means for the functions to be a basis.

It's definitely worth learning more about Fourier series and Fourier transforms if you have the time, but if this is the part that is confusing you, forget it for now -- "Fourier" is a red herring here. The important steps for computing the wave kernel are

1. Computing that the eigenfunctions $\mu$ of the Laplacian $f'' = \lambda f$ with periodic boundary conditions are precisely the sines and cosines listed above;
2. The eigenvalue corresponding to $\frac{1}{\sqrt{\pi}}\cos kx$ and $\frac{1}{\sqrt{\pi}}\sin kx$ is $-k^2$;
3. The wave kernel is given by taking the weighted sum of products of eigenvfunctions, as written in the formula in the other post.