Wiener amalgam space

Let ${\displaystyle X}$ be a normed space with norm ${\displaystyle \Vert \cdot {\Vert }_X}$. Then the Wiener amalgam space with local component ${\displaystyle X}$ and global component ${\displaystyle L_m^p}$, a weighted ${\displaystyle L^p}$ space with non-negative weight ${\displaystyle m}$, is defined by

${\displaystyle W(X,L^p)=\left\{f:{\left({\int }_{{\mathbb{R}}^d}\Vert f(\cdot )\overline{g}(\cdot -x){\Vert }_X^{p}m(x{)}^{p}dx\right)}^{1/p}<\mathrm{\infty }\right\},}$

where ${\displaystyle g}$ is a continuously differentiable, compactly supported function, such that ${\displaystyle \sum _{x\in {\mathbb{Z}}^{\mathbb{d}}}g(z-x)=1}$, for all ${\displaystyle z\in {\mathbb{R}}^d}$. Again, the space defined is independent of ${\displaystyle g}$. As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.