= (\mathbf{k}^{(1)}, \mathbf{k}^{(2)}, \mathbf{k}^{(3)})$, with components $\mathbf{k}^{(1)}\in \mathbb{N}_0^{s_1}$, $\mathbf{k}^{(2)}\in \mathbb{N}_0^{s_2}$, and $\mathbf{k}^{(3)}\in {\ensuremath{\mathbb{Z}}}^{s_3}$. The tensor product $\xi_{\mathbf{k}}= w_{\mathbf{k}^{(1)}} \otimes \gamma_{\mathbf{k}^{(2)}} \otimes e_{\mathbf{k}^{(3)}}$, where $w_{\mathbf{k}^{(1)}}\in \mathcal{W}_{\mathbf{b}^{(1)}}^{(s_1)}$, $\gamma_{\mathbf{k}^{(2)}}\in \Gamma_{\mathbf{b}^{(2)}}^{(s_2)}$, and $e_{\mathbf{k}^{(3)}}\in \mathcal{T}^{(s_3)}$, defines a function $\xi_{\mathbf{k}}$ on the $s$-dimensional unit cube, $$\xi_{\mathbf{k}}: [0,1)^s \rightarrow \mathbb{C},
\quad \xi_{\mathbf{k}}(\mathbf{x}) =
w_{\mathbf{k}^{(1)}}(\mathbf{x}^{(1)})  \gamma_{\mathbf{k}^{(2)}}(\mathbf{x}^{(2)})
    e_{\mathbf{k}^{(3)}}(\mathbf{x}^{(3)})\;,$$ where $\mathbf{x}= (\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \mathbf{x}^{(3)})\in [0,1)^s$.

Let $s_1, s_2, s_3\in {\ensuremath{\mathbb{N}}}_0$, not all $s_i$ equal to 0, and put $s=s_1+s_2+s_3$. The family of functions $$\mathcal{W}_{\mathbf{b}^{(1)}}^{(s_1)}\otimes \Gamma_{{\ensuremath{\mathbf b}}^{(2)}}^{(s_2)} \otimes \mathcal{T}^{(s_3)}=
    \{ \xi_{\ensuremath{\mathbf k}},
        {\ensuremath{\mathbf k}}= ({\ensuremath{\mathbf k}}^{(1)},{\ensuremath{\mathbf k}}^{(2)}, {\ensuremath{\mathbf k}}^{(3)})\in {\ensuremath{\mathbb{N}}}_0^{s_1}\times {\ensuremath{\mathbb{N}}}_0^{s_2}\times {\ensuremath{\mathbb{Z}}}^{s_3}\},$$ is called a [*hybrid function system*]{} on $[0,1)^s$.

It follows from [@Hel10c Theorem 1 and Corollary 4] and the techniques exhibited in [@Hel11a] that such hybrid function systems are an orthonormal basis of $L^2([0,1)^s)$.

The hybrid spectral test {#s:spectral}
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All of the following results remain valid if we change the order of the factors in the hybrid function system, as it will become apparent from the proofs below. In particular, out of the given $s$ coordinates, we may select arbitrary $s_1$ coordinates and assign to them the Walsh system $\mathcal{W}_{\mathbf{b}^{(1)}}^{(s_1)}$ in some base ${\ensuremath{\mathbf b}}^{(1)}$, treat $s_2$ of the remaining $s -s_1$ coordinates with a ${\ensuremath{\mathbf b}}^{(2)}$-adic system 