Dirichlet conditions for multivariate Fourier series.

Question

In my differential equations class, we saw that, if a periodic (resp. a periodic) function $f(x)$ satisfies the Dirichlet conditions, its Fourier series (resp. integral) converges pointwise to $f(x)$. We also defined multivariate Fourier series (for $f(x,y)$, $f(x,y,z)$, etc.), but we didn't define any multidimensional analogue of the Dirichlet conditions. What would this analogue be?