About mulit-variate Fourier series

Question

If $f(x,y)$ is $2\pi$ periodic with respect to $x$ and $2\pi$ periodic with respect to $y$ respectively, then can I write $$ f(x,y) = \sum_{j,k \in \mathbb Z} c_{jk} e^{ijx} e^{iky}$$ where $$ c_{jk} = \frac{1}{4\pi^2}\int_0^{2 \pi} \int_0^{2 \pi} f(x,y) e^{-ijx} e^{-iky} dxdy \;?$$ I am wondering the sufficient condition to write in this way. Is the condition $f(x + 2\pi, \cdot) = f(x, \cdot)$ and $f(\cdot, y + 2 \pi) = f(\cdot, y)$ sufficient? Or do I need $f(x + 2 \pi, y+2\pi) = f(x,y)$ ?

Answer 1

The conditions that $f$ is $2\pi$ periodic in the $x$ and $y$ directions separately imply together that $f(x + 2\pi, y + 2\pi) = f(x,y)$. Periodicity in both directions is needed.

However, there is also another condition that is required for the above to be true, which is that $f$ must be square-integrable; that is, $ \int_0^{2\pi} \int_0^{2\pi} |f(x,y)|^2 \, dx \, dy$ must be defined and finite.