if, for any sequence $\omega$ in $[0,1)^s$, the property $$\forall {\ensuremath{\mathbf k}}\in \Lambda^*: \lim_{N\to \infty} S_N(\xi_{\ensuremath{\mathbf k}}, \omega) =0$$ implies the uniform distribution of $\omega$ in $[0,1)^s$.

Examples of u.d.d. classes of functions on $[0,1)^s$ are the hybrid function systems $\mathcal{W}_{\mathbf{b}^{(1)}}^{(s_1)}\otimes \Gamma_{{\ensuremath{\mathbf b}}^{(2)}}^{(s_2)} \otimes \mathcal{T}^{(s_3)}$. This follows from the hybrid Weyl Criterion [@Hel10c Theorem 1], whose proof is easily adapted to provide for the non-prime bases $b_i$ we allow here in the factor $\Gamma_{{\ensuremath{\mathbf b}}^{(2)}}^{(s_2)}$.

\[def:weightfunction\] Let $||\cdot||$ be an arbitrary norm on ${\ensuremath{\mathbb{R}}}^s$ and let $\Lambda$ be an $s$-dimensional index set. We call a real-valued function $\rho$ a *weight function* on $\Lambda$ if, for all ${\ensuremath{\mathbf k}}\in \Lambda$, $\rho(\mathbf{k}) > 0$, and if for all $\epsilon>0$, there exists a positive real number $K_0=K_0(\epsilon)$ such that $\rho({\ensuremath{\mathbf k}})<\epsilon$ for all ${\ensuremath{\mathbf k}}\in\Lambda$ with $||{\ensuremath{\mathbf k}}||>K_0$.

With a u.d.d. function system ${\mathcal{F}}=\{ \xi_{\ensuremath{\mathbf k}}: {\ensuremath{\mathbf k}}\in \Lambda  \}$ on $[0,1)^s$ and a weight function $\rho$ on $\Lambda$ we may associate the array of weighted functions $$\label{s:spectral:eqn:array}
 \left( \rho({\ensuremath{\mathbf k}}) \xi_{\ensuremath{\mathbf k}}\right)_{{\ensuremath{\mathbf k}}\in \Lambda}.$$

The operator $S_N(\cdot, \omega)$ is linear, hence we may write $$S_N( \left( \rho({\ensuremath{\mathbf k}}) \xi_{\ensuremath{\mathbf k}}\right)_{{\ensuremath{\mathbf k}}\in \Lambda}, \omega)
    = \left( \rho({\ensuremath{\mathbf k}}) S_N(\xi_{\ensuremath{\mathbf k}}, \omega) \right)_{{\ensuremath{\mathbf k}}\in 