Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients.
I want to know if this result can be extended to higher dimensions. For example, consider $L^2[0,1]^2$, does it have an orthonormal basis as trigonometric series, possibly a tensor product forms, i.e. $\sin(nx)\cos(my)$, etc.
The motivation behind is that a common example to demonstrate spectral method is 1D heat equation, using trigonometric functions. On the other hand, 2D heat equation can also be solved using Fourier series, but I have not found any references discussing trigonometric orthonormal basis in higher dimensions. Besides, I am also interested in general orthonormal basis for $L^2[a,b]^d$ with $d>1$ and $W^{k,p}(\Omega)$, with $\Omega$ possibly bounded. I think Sobolev space needs to be taken care of more carefully with the regularity of $\partial \Omega$ (probably I should start a new question on this).