Wiener amalgam space

In mathematics, amalgam spaces categorize functions with regards to their local and global behavior. While the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgams, as the term is used today, were introduced by Hans Georg Feichtinger in 1980. The concept is named after Norbert Wiener.

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.

References

^ Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis by Hans Georg Feichtinger and Franz Luef

‹The stub template below has been proposed for renaming to . See stub types for deletion to help reach a consensus on what to do.
Feel free to edit the template, but the template must not be blanked, and this notice must not be removed, until the discussion is closed. For more information, read the guide to deletion.›