Finding a complete and orthonormal set of functions in the subspace of $L^2(-a, a)$ of the functions with mean value 0?

Question

Let said subspace be \begin{equation} \mathcal{F} =\{f \in L^2(-a,a) : \int_{-a}^af(x)dx=0\} \end{equation} What I need to do is find an orthonormal complete set of functions for this Hilbert subspace (I already know it's Hilbert). What I thought is of looking for a subset of the typical complete set of $L^2(-a,a) $, namely $\{sin(kx), cos(kx), k\geq 0 \text{integer} \}$, which is already orthonormal. The sines are odd functions and thus have mean value 0, so they should be part of the complete set I am looking for. The cosine with $ k=0 $ surely isn't part of this set (it's a constant!!). I am having trouble with all the other cosines though, depending on $a$, their mean value could be different from 0. Moreover, this reasoning I'm making is hardly a proof, but I'm kinda lost right now

Answer 1

For $a=\pi$ such a set is $\frac 1 {2\pi}e^{inx}, n \neq 0$ and you can make a simple substitution to get the answer for any $a$. [ Every $L^{2}$ function has a unique $L^{2}$ expansion if its F.S. $\sum a_ne^{inx}$ and the integral of $f$ is $0$ iff the constant term vanishes].