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In mathematics, a function on a normed vector space is said to vanish at infinity if

f(x)→0{displaystyle f(x)to 0} as ‖x‖→∞.{displaystyle |x|to infty .}

For example, the function

f(x)=1x2+1{displaystyle f(x)={frac {1}{x^{2}+1}}}

defined on the real line vanishes at infinity.

More generally, a function f{displaystyle f} on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number ϵ{displaystyle epsilon }, there is a compact subset K{displaystyle K} such that

‖f(x)‖<ϵ{displaystyle |f(x)|<epsilon }

whenever the point x{displaystyle x} lies outside of K{displaystyle K}.

In other words, for each positive number ϵ{displaystyle epsilon } the set
{x∈X:‖f(x)‖≥ϵ}{displaystyle left{xin X:|f(x)|geq epsilon right}} is compact.

For a given locally compact space Ω{displaystyle Omega }, the set of such functions

f:Ω→K{displaystyle f:Omega rightarrow mathbb {K} }

(where K{displaystyle mathbb {K} } is either the field R{displaystyle mathbb {R} } of real numbers or the field C{displaystyle mathbb {C} } of complex numbers) forms an K{displaystyle mathbb {K} }-vector space with respect to pointwise scalar multiplication and addition, often denoted C0(Ω){displaystyle C_{0}(Omega )}.

(Warning: These definitions are inconsistent. If f(x)=‖x‖−1{displaystyle f(x)=|x|^{-1}} in an infinite dimensional Banach
space, then f{displaystyle f} vanishes at infinity by the ‖f(x)‖→0{displaystyle |f(x)|to 0} definition but not by the compact set definition.)

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 point at infinity.

Rapidly decreasing

Refining the concept, one can look more closely to the rate of vanishing of functions at infinity. One of the basic intuitions of mathematical analysis is that the Fourier transform interchanges smoothness conditions with rate conditions on vanishing at infinity. The rapidly decreasing test functions of tempered distribution theory are smooth functions that are

o(|x|^−N)

for all N, as |x| → ∞, and such that all their partial derivatives satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding distribution theory of tempered distributions will have the same good property.

References

• 
  Hewitt, E and Stromberg, K (1963). Real and abstract analysis. Springer-Verlag.CS1 maint: Multiple names: authors list (link) .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}