\cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j).$$

We usually assume that $\theta _1= \cdots =\theta _d={\varepsilon}>0$, and then we set $\Lambda _{\varepsilon}= \Lambda _{[\theta ]}$.

The proof of the following result is omitted since the result follows from Theorem 13.1.1 and other results in [@Gc2] (see also Theorem S in [@Gc1]).

\[ThmS\] Let $\Lambda$ be as in , $v\in {\mathscr P}_E({\mathbf R^{2d}})$ be submultiplicative, and $\phi \in
M^1_{(v)}({\mathbf R^{d}})\setminus 0$. Then the following is true:

1.  if $$\label{DualFrames}
    \{ e^{i{\langle {\, \cdot \, },\iota \rangle}}\phi ({\, \cdot \, }-j) \} _{j,\iota \in \Lambda}
    \quad \text{and}\quad
    \{ e^{i{\langle {\, \cdot \, },\iota \rangle}}\psi ({\, \cdot \, }-j) \} _{j,\iota \in \Lambda}$$ are dual frames to each others, then $\psi \in M^1_{(v)}({\mathbf R^{d}})$;

2.  there is a constant ${\varepsilon}_0>0$ such that the frame operator $S_{\phi ,\phi}^\Lambda$ is a homeomorphism on $M^1_{(v)}({\mathbf R^{d}})$ and are dual frames, when $\Lambda ={\varepsilon}{\mathbf Z^{d}}$, ${\varepsilon}\in (0,{\varepsilon}_0]$ and $\psi = (S_{\phi ,\phi}^\Lambda )^{-1}\phi$.

We also recall the following restatement of [@Toft12 Theorem 3.7] (see also Corollaries 12.2.5 and 12.2.6 in [@Gc2] and Theorem 3.7 in [@GaSa]). Here and in what follows we let $\Lambda ^2=\Lambda \times \Lambda$.

\[ConseqThmS\] Let $\Lambda$ be as in , ${{\boldsymbol p}}\in
