base $b_i$, $1\le i\le s$. The Walsh function system in base $\mathbf{b}$ in dimension $s$ is denoted by $\mathcal{W}_{\mathbf{b}}^{(s)} = \{w_{\mathbf{k}}:   \mathbf{k}\in \mathbb{N}_0^s\}$.

The trigonometric function system defined below is the classical function system in the theory of uniform distribution of sequences (see the monograph [@Kui74a]).

Let $k\in \mathbb{Z}$. The $k$th [*trigonometric function*]{} $e_k$ is defined as $ e_k: [0,1) \rightarrow \mathbb{C}$, $ e_k(x)
= e(kx)$. For ${\ensuremath{\mathbf k}}=(k_1, \ldots, k_s)\in \mathbb{Z}^s$, the [*${\ensuremath{\mathbf k}}$th trigonometric function*]{} $e_{{\ensuremath{\mathbf k}}}$ is defined as $e_{{\ensuremath{\mathbf k}}}: [0,1)^s \rightarrow \mathbb{C}$, $e_{{\ensuremath{\mathbf k}}}({{\ensuremath{\mathbf x}}}) =
\prod_{i=1}^s e(k_ix_i)$, ${\ensuremath{\mathbf x}}=(x_1, \ldots, x_s)\in [0,1)^s$. The [*trigonometric function system*]{} in dimension $s$ is denoted by $\mathcal{T}^{(s)} = \{e_{{\ensuremath{\mathbf k}}}: {\ensuremath{\mathbf k}}\in
\mathbb{Z}^s\}$.

The following presentation complements the concepts discussed in [@Hel10c]. As will become clear, any finite number of factors can be accomodated. For given dimensions $s_1, s_2$ and $s_3$, with $s_i \in \mathbb{N}_0$, not all equal to 0, put $s=s_1+s_2+s_3$ and write a point ${\ensuremath{\mathbf y}}\in {\ensuremath{\mathbb{R}}}^s$ in the form ${\ensuremath{\mathbf y}}=({\ensuremath{\mathbf y}}^{(1)}, {\ensuremath{\mathbf y}}^{(2)}, {\ensuremath{\mathbf y}}^{(3)})$ with components ${\ensuremath{\mathbf y}}^{(j)}\in {\ensuremath{\mathbb{R}}}^{s_j}$, $j=1,2,3$. Let us fix two vectors of bases $\mathbf{b}^{(1)}=(b_1, \ldots, b_{s_1})$, and $\mathbf{b}^{(2)}=(b_{s_1+1}, \ldots, b_{s_1+s_2})$ with not necessarily distinct integers $b_i\ge 2$. Let $\mathbf{k} 