Vanish at infinity

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

as

For example, the function

defined on the real line vanishes at infinity. The same applies to the function

where and are real and correspond to the point on .[1]

More generally, a function on a locally compact space (which may or may not have a norm) vanishes at infinity, if given any positive number , there exists a compact subset such that

whenever the point lies outside of .[2][3][4]

In other words, for each positive number the setis compact.
For a given locally compact space , the set of such functions

(where is either or ) forms an -vector space with respect to pointwise scalar multiplication and addition, often denoted .

Here, note that the two definitions could be inconsistent with each other: if in an infinite dimensional Banach space, then vanishes at infinity by the definition, but not by the compact set definition.

Aside from this difference, 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 one approaches it. This definition can be formalized in many cases by adding an (actual) point at infinity.

Rapidly decreasing[edit]

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

for all , as , and such that all their partial derivatives satisfy the same 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.

See also[edit]

References[edit]

  1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Vanish". Math Vault. 2019-08-01. Retrieved 2019-12-15.
  2. ^ "Function vanishing at infinity - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-15.
  3. ^ "vanishing at infinity in nLab". ncatlab.org. Retrieved 2019-12-15.
  4. ^ "vanish at infinity". planetmath.org. Retrieved 2019-12-15.

Bibliography[edit]

  • Hewitt, E and Stromberg, K (1963). Real and abstract analysis. Springer-Verlag.CS1 maint: multiple names: authors list (link)