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  "body": "\n## Definition\n\nLet $d \\ge 1$, let $1 \\le p, q \\le \\infty$ and let $Q = [0, 1)^d$.\nFor a measurable function $f \\colon \\R^d \\to \\C$ and $k \\in \\Z^d$, let $a_k(f) = \\norm{f \\cdot 1_{k + Q}}_{L^p(\\R^d)}$, the norm in the [[Lebesgue space|SASP14MA1Z]] $L^p$ of the restriction of $f$ to the cube $k + Q$.\nThe *Wiener amalgam space* $W(L^p, \\ell^q)(\\R^d)$ is the set of measurable $f$ with\n$$\\norm{f}_{W(L^p, \\ell^q)} = \\Big( \\sum_{k \\in \\Z^d} a_k(f)^q \\Big)^{1/q} \u003c \\infty,$$\nwith the supremum over $k$ in place of the sum when $q = \\infty$.\nFunctions that agree almost everywhere are identified.\n\n## Remarks\n\n$L^p$ is the local component and $\\ell^q$ the global component.\nIn the general theory of Feichtinger, $L^p$ is replaced by a Banach space $B$ of local behaviour.\nThe 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.\nTwo facts follow from the definition:\n\n1. $W(L^p, \\ell^p) = L^p(\\R^d)$ with equal norms.\n2. 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 $1$.\n\nExample: let $d = 1$ and $f(x) = 1/(1 + \\abs{x})$, with $p = \\infty$.\nOn $[k, k+1)$ the supremum of $f$ is $1/(1 + \\min(\\abs{k}, \\abs{k + 1}))$.\nThese numbers behave like $1/\\abs{k}$, so their $q$-th powers have a finite sum for every $q \u003e 1$.\nHence $f \\in W(L^\\infty, \\ell^q)(\\R)$ for $q \u003e 1$.\n\nNon-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.\n",
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