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| In [[mathematics]], a [[function (mathematics)|function]] on a [[normed vector space]] is said to '''vanish at infinity''' if
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| :<math>f(x)\to 0</math> as <math>\|x\|\to \infty.</math>
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| For example, the function
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| :<math>f(x)=\frac{1}{x^2+1}</math> | |
| defined on the [[real line]] vanishes at infinity.
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| More generally, a function <math>f</math> on a [[locally compact space]] (which may not have a norm) vanishes at infinity if, given any [[positive number]] <math>\epsilon</math>, there is a [[compact space|compact]] subset <math>K</math> such that
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| :<math>\|f(x)\| < \epsilon</math> | |
| whenever the point <math>x</math> lies outside of <math>K</math>. For a given [[locally compact]] space <math>\Omega</math>, the [[set (mathematics)|set]] of such functions
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| :<math>f:\Omega\rightarrow\mathbb{F}</math> | |
| (where <math>\mathbb{F}</math> is either the [[field (mathematics)|field]] <math>\mathbb{R}</math> of [[real number]]s or the field <math>\mathbb{C}</math> of [[complex number]]s) forms an <math>\mathbb{F}</math>-vector space with respect to [[pointwise]] [[scalar multiplication]] and [[addition]], often denoted <math>C_{0}(\Omega)</math>.
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| Both of these notions correspond to the intuitive notion of adding a point "at infinity" and requiring the values of the function to get arbitrarily close to zero as we approach it. This "definition" can be formalized in many cases by adding a [[locally compact#The point at infinity|point at infinity]].
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| ==Rapidly decreasing==
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| Refining the concept, one can look more closely at the ''rate of vanishing'' of functions at infinity. One of the basic intuitions of [[mathematical analysis]] is that the [[Fourier transform]] interchanges [[smooth function|smoothness]] conditions with rate conditions on vanishing at infinity. The '''rapidly decreasing''' test functions of [[tempered distribution]] theory are [[smooth function]]s that are
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| :o(|''x''|<sup>−''N''</sup>)
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| for all ''N'', as |''x''| → ∞, and such that all their [[partial derivative]]s satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding [[Distribution (mathematics)|distribution theory]] of ''tempered distributions'' will have the same good property.
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| ==See also==
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| * [[Schwartz space]]
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| ==References==
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| {{Refimprove|date=January 2008}}
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| * {{cite book|author=Hewitt, E and Stromberg, K|year=1963|title=Real and abstract analysis|publisher=Springer-Verlag}}
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| [[Category:Mathematical analysis]]
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