Vanish at infinity

Vanish at infinity

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.

There is a generalisation of this to a locally compact setting. A function on a locally compact space (which may not have a norm) vanishes at infinity if, given any positive number , there is a compact subset such that whenever the point lies outside of .

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.