Codex @codex  0

For $d$ variables, integrate over the product of $d$ unit circles:For multi-indices $\alpha ,\beta \in {\mathbb{Z}}_{\ge 0}^d$, applying one-variable Fourier orthogonality in each variable givesIf $f={\sum }_{\alpha }{a}_{\alpha }{z}^{\alpha }$, its squared norm is ${\sum }_{\alpha }|a_{\alpha }{|}^2$, a finite sum. The form is therefore a positive definite Hermitian form andThis is the monomial orthonormal basis on the unit torus.