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{{Notability|date=June 2011}} | |||
In mathematics, '''amalgam spaces''' categorize functions with regard to their local and global behavior. While the concept of [[function space]]s 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 <math> X </math> be a normed space with norm <math> \|\cdot \|_X </math>. Then the ''Wiener amalgam space''<ref>Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis by Hans Georg Feichtinger and Franz Luef</ref> with local component <math> X </math> and | |||
global component <math> L^p_m </math>, a [[Lp_space#Weighted_Lp_spaces|weighted <math>L^p</math> space]] with non-negative weight <math> m </math>, is defined by | |||
:<math> 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\}, </math> | |||
where <math> g </math> is a continuously differentiable, compactly supported function, such that <math> \sum_{x\in\mathbb{Z^d}} g(z-x) = 1 </math>, for all <math> z\in\mathbb{R}^d </math>. Again, the space defined is independent of <math> g </math>. As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.<ref>Foundations of Time-Frequency Analysis by Karlheinz Gröchenig</ref> | |||
==References== | |||
{{reflist}} | |||
[[Category:Function spaces]] | |||
{{mathanalysis-stub}} |
Revision as of 11:32, 25 January 2014
Template:Notability In mathematics, amalgam spaces categorize functions with regard 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 be a normed space with norm . Then the Wiener amalgam space[1] with local component and global component , a weighted space with non-negative weight , is defined by
where is a continuously differentiable, compactly supported function, such that , for all . Again, the space defined is independent of . As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.[2]
References
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