Wiener amalgam space

Remarks

$L^p$ is the local component and $\ell^q$ the global component. In the general theory of Feichtinger, $L^p$ is replaced by a Banach space $B$ of local behaviour. The space $\ell^q$ is replaced by a solid sequence or function space $C$ of global behaviour, and the sharp cut-off $1_{k + Q}$ by a smooth partition of unity. Two facts follow from the definition:

1. $W(L^p, \ell^p) = L^p(\R^d)$ with equal norms.
2. If $p_1 \ge p_2$ and $q_1 \le q_2$, then $W(L^{p_1}, \ell^{q_1}) \subseteq W(L^{p_2}, \ell^{q_2})$, since the cube $Q$ has measure $1$.

Example: let $d = 1$ and $f(x) = 1/(1 + \abs{x})$, with $p = \infty$. On $[k, k+1)$ the supremum of $f$ is $1/(1 + \min(\abs{k}, \abs{k + 1}))$. These numbers behave like $1/\abs{k}$, so their $q$-th powers have a finite sum for every $q > 1$. Hence $f \in W(L^\infty, \ell^q)(\R)$ for $q > 1$.

Non-example: the same $f$ is not in $W(L^\infty, \ell^1)(\R)$, because the sum of these numbers is a harmonic series and diverges.