Let's say you have a function that is periodic in only one direction (x-direction), given by:
$$ f(x,y) $$ $$ 0 \leq x \leq L $$
If smoothness and continuity of the function in all directions is allowed to be assumed, also, let's assume the function has odd periodicity in the x-direction then, is it feasible to write the coefficient for the Fourier series as a function of y? In particular, is it feasible to write the function given by the below: $$ f(x,y) = \sum_{n} a_{n}(y)\sin(\frac{n\pi x}{L}) $$
Assuming the usual integration is done to find the coefficient of the series. If this is allowed, when would it be interesting to write a function this way, if at all?