Vanish at infinity

In mathematics, a function on a normed vector space is said to vanish at infinity if Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)\to 0

as Failed to parse (Missing texvc executable; please see math/README to configure.): \|x\|\to \infty.
For example, the function 

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=\frac{1}{1+x^2}

defined on the real line vanishes at infinity.

There is a generalization of this to a locally compact setting. A function Failed to parse (Missing texvc executable; please see math/README to configure.): f

on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon

, there is a compact subset Failed to parse (Missing texvc executable; please see math/README to configure.): K

such that Failed to parse (Missing texvc executable; please see math/README to configure.): \|f(x)\| < \epsilon
whenever the point Failed to parse (Missing texvc executable; please see math/README to configure.): x
lies outside of Failed to parse (Missing texvc executable; please see math/README to configure.): K

.

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 at 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.