\Lambda}.$$

\[definition:generalspectraltest\] Let ${\mathcal{F}}=\{ \xi_{\ensuremath{\mathbf k}}: {\ensuremath{\mathbf k}}\in \Lambda  \}$ be a u.d.d. function system on $[0,1)^s$, and let $\rho$ be a weight function on $\Lambda$. For a given sequence $\omega = (\mathbf{x}_n)_{n\ge 0}$ in $[0,1)^s$, the *spectral test* $\sigma_N(\omega)$ of the first $N$ elements of $\omega$, with respect to ${\mathcal{F}}$ and $\rho$, is defined as $$\begin{aligned}
\sigma_N(\omega) &= || \left( \rho({\ensuremath{\mathbf k}})  \right)_{{\ensuremath{\mathbf k}}\in \Lambda^*}||_\infty^{-1}\
    || \left( \rho({\ensuremath{\mathbf k}}) S_N(\xi_{\ensuremath{\mathbf k}}, \omega) \right)_{{\ensuremath{\mathbf k}}\in \Lambda^*} ||_\infty \\
    &= \sup_{{\ensuremath{\mathbf k}}\in \Lambda^*} \{ \rho({\ensuremath{\mathbf k}}) \}^{-1}
    \sup_{{\ensuremath{\mathbf k}}\in \Lambda^*} \left\{
        \rho({\ensuremath{\mathbf k}}) \left| S_N( \xi_\mathbf{k}, \omega) \right| \right\}.\end{aligned}$$ Let $\alpha>1$ be a given real number. If the weight function $\rho$ fulfills the additional condition $$\sum_{{\ensuremath{\mathbf k}}\in \Lambda} \rho({\ensuremath{\mathbf k}})^\alpha < \infty,$$ then the $L^\alpha$-*diaphony* $F_N^{(\alpha)}(\omega)$ of the first $N$ elements of $\omega$, with respect to ${\mathcal{F}}$ and $\rho$, is defined as $$\begin{aligned}
F_N^{(\alpha)}(\omega) &= ||  \left( \rho({\ensuremath{\mathbf k}}) \right)_{{\ensuremath{\mathbf k}}\in \Lambda^*}||_\alpha^{-1}
    \ || \left( \rho({\ensuremath{\mathbf k}}) S_N(\xi_{\ensuremath{\mathbf k}}, \omega) \right)_{{\ensuremath{\mathbf k}}\in \Lambda^*} ||_\alpha\\
   &= \left( \sum_{{\ensuremath{\mathbf k}}\in \Lambda^*} \rho({\ensuremath{\mathbf k}})^\alpha\right)^{-1/\alpha}\
    \left( \sum_{{\ensuremath{\mathbf k}}\in \Lambda^*} \rho({\ensuremath{\mathbf k}})^\alpha \left| S_N( \xi_\mathbf{k}, \omega) \right|^\alpha\right)^{1/\alpha}.\end{aligned}$$

We note that the fact $|S_N(\xi_{\ensuremath{\mathbf k}}, \omega)|\le 1$ implies that the spectral test as well as diaphony are 