In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find a orthonormal basis $\{e_n\}_{n\in {\bf Z}}$ for $L^2(\Omega)$ where $e_n(x)=\frac{1}{\sqrt{2\pi}}e^{inx}.$
In the high dimension, say, $\Omega\subseteq {\bf R}^n$ and $\Omega$ being bounded, can one still "construct" the orthonormal basis?