If I have some function $V(x,y)$ which is periodic in x with period L. I wish to expand $V(x,y)$ in terms of a fourier sine (for simplicity) series in $x$, is it always the case that I may write the following?
$$V(x,y) = \sum_{n = 1}^{\infty} a_n \sin(\frac{n\pi x}{L})f_n(y)$$
Where the $f_n(y)$ are whatver functions of $y$ needed to satisfy the equality -- do they need to have a particular form or any properties other than continuity?
It seems to me that we might require a certain type of function $V(x,y)$ to do this, does anyone know what the neccessary and sufficient conditions on $V(x,y)$ are to be able to expand in this way?
Not really. Your expansion will, rather, take the form
$$V(x,y) = \sum_{n=1}^{\infty} \: \sum_{m=1}^{\infty} a_{nm} \sin{\left ( \frac{n \pi x}{L}\right)} f_m(y)$$
$f_m$ will be defined by boundary conditions on the $y$ boundaries.