$(S_{\phi ,\phi}^\Lambda )^{-1}\phi$ is called the *canonical dual window of $\phi$*, with respect to $\Lambda$. We have $$S_{\phi ,\phi}^\Lambda (e^{i{\langle {\, \cdot \, },\iota \rangle}}f({\, \cdot \, }-j)) =
e^{i{\langle {\, \cdot \, },\iota \rangle}}(S_{\phi ,\phi}^\Lambda f)({\, \cdot \, }-j),$$ when $f\in M^\infty _{(1/v)}({\mathbf R^{d}})$ and $j,\iota \in \Lambda$. The series in are called *Gabor expansions of $f$* with respect to $\phi$ and $\psi$.

\[RemThmS\] There are several ways to achieve dual frames satisfying the required properties in Proposition \[ConseqThmS\]. In fact, let $v,v_{0}\in {\mathscr P}_E({\mathbf R^{2d}})$ be submultiplicative such that $\omega $ is $v$-moderate and $L^1_{(v_0)}({\mathbf R^{2d}})\subseteq
L^r({\mathbf R^{2d}})$. Then Proposition \[ThmS\] guarantees that for some choice of $\phi ,\psi \in M^1_{(v_0v)}({\mathbf R^{d}})\subseteq
M^r_{(v)}({\mathbf R^{d}})$ and lattice $\Lambda$ in , the sets in are dual frames to each others, and that $\psi = (S_{\phi ,\phi}^\Lambda )^{-1}\phi$.

In the sequel we usually assume that $\Lambda =\Lambda _{\varepsilon}$, with ${\varepsilon}>0$ small enough such that the hypotheses in Propositions \[ThmS\] and \[ConseqThmS\] are fulfilled, and that the window functions and their duals belong to $M^r_{(v)}$ for every $r>0$. This is always possible, in view of Remark \[RemThmS\].

Classes of matrices {#subsec1.5}
-------------------

In what follows we let $\Lambda$ be a in , 