and let ${\mathbf{k}}= (k_1, \ldots, k_s)\in \mathbb{N}_0^s$. We define $\varphi_{\mathbf{b}}(\mathbf{z}) = (\varphi_{b_1}(z_1), \ldots, \varphi_{b_s}(z_s))$, and $\varphi^+_{\mathbf{b}}(\mathbf{x}) = (\varphi^+_{b_1}(x_1), \ldots, \varphi^+_{b_s}(x_s))$.

Let $\chi_{\mathbf{k}}(\mathbf{z}) = \prod_{i=1}^s \chi_{k_i}(z_i)$, where $\chi_{k_i} \in \hat{\mathbb{Z}}_{b_i}$, and define $\gamma_{\mathbf{k}}({\mathbf{x}}) = \prod_{i=1}^s \gamma_{k_i}(x_i)$, where $\gamma_{k_i} \in \Gamma_{b_i}$, $1\le i\le s$. Then $\gamma_{\mathbf{k}} = \chi_{\mathbf{k}}\circ \varphi^+_{\mathbf{b}}$. Let $\Gamma_{\mathbf{b}}^{(s)} = \{ \gamma_{\mathbf{k}}:
\mathbf{k} \in \mathbb{N}_0^s \}$ denote the [*$\mathbf{b}$-adic function system*]{} in dimension $s$.

The dual group $\hat{{\ensuremath{\mathbb{Z}}}}_{\ensuremath{\mathbf b}}$ is an orthonormal basis of the Hilbert space $L^2({\ensuremath{\mathbb{Z}}}_{\ensuremath{\mathbf b}})$. A rather elementary proof of this result is given in [@Hel11a Theorem 2.12].

For the Walsh functions defined below, we refer the reader to [@Dick10a; @Hel93a; @Hel98a] for elementary properties of these functions and to [@Sch90a] for the background in harmonic analysis.

For $k\in \mathbb{N}_0$, $k= \sum_{j\ge 0} k_j b^j$, and $x\in [0,1)$, with regular $b$-adic representation $x= \sum_{j\ge 0} x_j b^{-j-1}$, the [*$k$th Walsh function in base $b$*]{} is defined by $w_k(x) = e( (\sum_{j\ge 0} k_j x_j)/b)$. For ${\mathbf{k}}\in \mathbb{N}_0^s$, ${\mathbf{k}}=(k_1, \ldots, k_s)$, and ${\mathbf{x}}\in [0,1)^s$, ${\mathbf{x}}=(x_1, \ldots, x_s)$, we define the [*${\mathbf{k}}$th Walsh function $w_{\mathbf{k}}$ in base $\mathbf{b}=(b_1, \ldots, b_s)$*]{} on $[0,1)^s$ as the following product: $w_{\mathbf{k}}(\mathbf{x}) = \prod_{i=1}^s
    w_{k_i}(x_i)$, where $w_{k_i}$ denotes the $k_i$th Walsh function in 