\in \Lambda$, where $$\label{LambdaDef}
\begin{aligned}
\Lambda &= \Lambda _{[\theta ]} = T_\theta {\mathbf Z^{d}} \equiv
{\{ \, (\theta _1j_1,\dots ,\theta _dj_d)\, ;\, (j_1,\dots ,j_d)\in {\mathbf Z^{d}}\, \} } ,
\\[1ex]
\theta &=(\theta _1,\dots ,\theta _d)\in {\mathbf R_*^{d}},
\end{aligned}$$ and $T_\theta$ denotes the diagonal matrix with diagonal elements $\theta _1,\dots ,\theta _d$. In the former case we set $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu)=L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}=
L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}({\mathbf R^{d}})$, and in the latter case we set $L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\mu) =
\ell ^{{{\boldsymbol p}}}_{\sigma ,(\omega )}(\Lambda )$.

For conveniency we also set $L^{{{\boldsymbol p}}}_{(\omega )}=L^{{{\boldsymbol p}}}_{\sigma ,(\omega )}$ and $\ell ^{{{\boldsymbol p}}} _{(\omega )}=\ell ^{{{\boldsymbol p}}} _{\sigma ,(\omega )}$ when $\sigma$ is the identity map, and we let $\ell (\Lambda )$ be the set of all (complex-valued) sequences on $\Lambda$ and $\ell _0
(\Lambda )$ be the set of all $f\in \ell (\Lambda )$ such that $f(j)\neq 0$ for at most finite numbers of $j$. Furthermore, if $\omega$ is equal to $1$, then we set $$L^{{{\boldsymbol p}}}_{\sigma } =L^{{{\boldsymbol p}}}_{\sigma ,(\omega )},
\quad
\ell ^{{{\boldsymbol p}}} _{\sigma }=\ell ^{{{\boldsymbol p}}} _{\sigma ,(\omega )},
\quad
L^{{{\boldsymbol p}}} = L^{{{\boldsymbol p}}}_{(\omega )}
\quad \text{and}  \quad
\ell ^{{{\boldsymbol p}}} = \ell ^{{{\boldsymbol p}}} 