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In [[functional analysis]], a '''weighted space''' is a space of functions under a ''weighted norm'', which is a finite [[norm (mathematics)|norm]] (or semi-norm) that involves multiplication by a particular function referred to as the ''weight''.


Weights can be used to expand or reduce a space of considered functions. For example, in the space of functions from a set <math>U\subset\mathbb{R}</math> to <math>\mathbb{R}</math> under the norm <math>\|\cdot\|_U</math> defined by: <math>\|f\|_U=\sup_{x\in U}{|f(x)|}</math>, functions that have infinity as a [[limit point]] are excluded. However, the weighted norm <math>\|f\|=\sup_{x\in U}{\left|f(x)\tfrac{1}{1+x^2}\right|}</math> is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm <math>\|f\|=\sup_{x\in U}{\left|f(x)x^4\right|}</math> is finite for many fewer functions.


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When the weight is of the form <math>\tfrac{1}{1+x^m}</math>, the weighted space is called ''polynomial-weighted''.<ref>{{Cite journal|last=Walczak|first=Zbigniew|year=2005|title=On the rate of convergence for some linear operators|journal=[[Hiroshima Mathematical Journal]]|volume=35|pages=115–124|url=http://www.math.sci.hiroshima-u.ac.jp/hmj/v35.1/p115-124.pdf}}</ref>
 
==References==
<references />
*{{Cite encyclopedia|editor=[[Michiel Hazewinkel]]|year=2001|encyclopedia=Encyclopaedia of Mathematics|publisher=Springer|last=Kudryavtsev|first=L&#x200A;D|title=Weighted Space|url=http://eom.springer.de/W/w097580.htm}}
 
[[Category:Functional analysis]]
 
 
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Latest revision as of 08:21, 8 January 2014

My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.

Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) In functional analysis, a weighted space is a space of functions under a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the weight.

Weights can be used to expand or reduce a space of considered functions. For example, in the space of functions from a set U to under the norm U defined by: fU=supxU|f(x)|, functions that have infinity as a limit point are excluded. However, the weighted norm f=supxU|f(x)11+x2| is finite for many more functions, so the associated space contains more functions. Alternatively, the weighted norm f=supxU|f(x)x4| is finite for many fewer functions.

When the weight is of the form 11+xm, the weighted space is called polynomial-weighted.[1]

References

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