Vanish at infinity
Encyclopedia
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.
More generally, 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 
. For a given locally compact space 
, the set of functions
(where
is either the field
of real number
s or the field
of complex number
s) forms an
-vector space with respect to pointwise
scalar multiplication
and addition
, often denoted
.
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.
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 function
s that are
for all N, as |x| → ∞, and such that all their partial derivative
s 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
on a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....
is said to vanish at infinity if
as For example, the function
defined on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
vanishes at infinity.
More generally, a function
on a locally compact spaceLocally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
(which may not have a norm) vanishes at infinity if, given any positive number
, there is a compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
subset
such thatwhenever the point
lies outside of . For a given locally compact space , the set of functions(where
is either the fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s or the field
of complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s) forms an
-vector space with respect to pointwisePointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations — operations defined on functions by applying the operations to function values...
scalar multiplication
Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...
and addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, often denoted
.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 at the rate of vanishing of functions at infinity. One of the basic intuitions of mathematical analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
is that the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
interchanges smoothness
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
conditions with rate conditions on vanishing at infinity. The rapidly decreasing test functions of tempered distribution
Tempered distribution
*Distribution *Tempered representation...
theory are smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
s that are
- o(|x|−N)
for all N, as |x| → ∞, and such that all their partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s satisfy that condition, too. This condition is set up so as to be self-dual under Fourier transform, so that the corresponding distribution theory
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
of tempered distributions will have the same good property.