_{(\omega )}.$$

Let ${{\boldsymbol p}}= (p_1,\dots ,p_d)\in [0,\infty ]^d$, ${\boldsymbol q}=(q_1,\dots ,q_d)\in [0 ,\infty ]^d$ and $t\in [-\infty ,\infty ]$. Then we use the conventions $${{\boldsymbol p}}\le {\boldsymbol q}\quad \text{and}\quad {{\boldsymbol p}}\le t \quad \text{when}\quad
p_j\le q_j\ \text{and}\ p_j\le t,$$ respectively, for every $j=1,\dots ,d$, and $${{\boldsymbol p}}= {\boldsymbol q}\quad \text{and}\quad {{\boldsymbol p}}= t \quad \text{when}\quad
p_j= q_j\ \text{and}\ p_j= t,$$ respectively, for every $j=1,\dots ,d$. The relations ${{\boldsymbol p}}< {\boldsymbol q}$ and ${{\boldsymbol p}}< t$ are defined analogously. We also let $${{\boldsymbol p}}\pm {\boldsymbol q}=(p_1\pm q_1,\dots ,p_d\pm q_d)
\quad \text{and}\quad
{{\boldsymbol p}}\pm t=(p_1\pm t,\dots ,p_d\pm t),$$ provided the right-hand sides are well-defined and belongs to $[-\infty ,\infty ]^d$. Moreover, we set $1/0=\infty$, $1/\infty =0$ and $1/{{\boldsymbol p}}=(1/p_1,\dots ,1/p_d)$.

We also let $$\max ({{\boldsymbol p}}) \equiv \max (p_1,\dots ,p_d)
\quad \text{and}\quad
\min ({{\boldsymbol p}}) \equiv \min (p_1,\dots ,p_d),$$ and note that if $\max ({{\boldsymbol p}}) <\infty$, then $\ell _0 (\Lambda )$ is dense in $\ell ^{{\boldsymbol p}}_{\sigma ,(\omega )} (\Lambda )$.

Modulation spaces {#subsec1.2}
-----------------

Next we define modulation spaces. Let $\phi \in {\mathcal S}_{1/2}({\mathbf R^{d}})\setminus 0$. For any $p,q\in (0.\infty ]$ and $\omega \in {\mathscr P}_E({\mathbf R^{2d}})$, the modulation spaces $M^{p,q}_{(\omega )}({\mathbf R^{d}})$ and $W^{p,q}_{(\omega )}({\mathbf R^{d}})$ are the 