# Wiener amalgam space

Let $X$ be a normed space with norm $\|\cdot \|_X$. Then the Wiener amalgam space[1] with local component $X$ and global component $L^p_m$, a weighted $L^p$ space with non-negative weight $m$, is defined by
$W(X,L^p) = \left\{ f\ :\ \left(\int_{\mathbb{R}^d} \|f(\cdot)\bar{g}(\cdot-x)\|^p_X m(x)^p \, dx\right)^{1/p} < \infty\right\},$
where $g$ is a continuously differentiable, compactly supported function, such that $\sum_{x\in\mathbb{Z^d}} g(z-x) = 1$, for all $z\in\mathbb{R}^d$. Again, the space defined is independent of $g$. As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.[2]