Vanish at infinity
For example, the function
defined on the real line vanishes at infinity.
(Warning: These definitions are inconsistent. If in an infinite dimensional Banach space, then vanishes at infinity by the 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.
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 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.
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- Hewitt, E and Stromberg, K (1963). Real and abstract analysis. Springer-Verlag.CS1 maint: multiple names: authors list (link)