In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a bump function (also called a test function) is a
function on a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
which is both
smooth (in the sense of having
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of all orders) and
compactly supported. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all bump functions with
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
forms a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, denoted
or
The
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of this space endowed with a suitable
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
is the space of
distributions.
Examples
The function
given by
is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded closed support. The proof of smoothness follows along the same lines as for the related function discussed in the
Non-analytic smooth function
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is ...
article. This function can be interpreted as the
Gaussian function scaled to fit into the unit disc: the substitution
corresponds to sending
to
A simple example of a (square) bump function in
variables is obtained by taking the product of
copies of the above bump function in one variable, so
Existence of bump functions
It is possible to construct bump functions "to specifications". Stated formally, if
is an arbitrary
compact set in
dimensions and
is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
containing
there exists a bump function
which is
on
and
outside of
Since
can be taken to be a very small neighborhood of
this amounts to being able to construct a function that is
on
and falls off rapidly to
outside of
while still being smooth.
The construction proceeds as follows. One considers a compact neighborhood
of
contained in
so
The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of
will be equal to
on
and
outside of
so in particular, it will be
on
and
outside of
This function is not smooth however. The key idea is to smooth
a bit, by taking the
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of
with a
mollifier. The latter is just a bump function with a very small support and whose integral is
Such a mollifier can be obtained, for example, by taking the bump function
from the previous section and performing appropriate scalings.
An alternative construction that does not involve convolution is now detailed.
Start with any smooth function
that vanishes on the negative reals and is positive on the positive reals (that is,
on
and
on
where continuity from the left necessitates
); an example of such a function is
for
and
otherwise.
Fix an open subset
of
and denote the usual
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
by
(so
is endowed with the usual
Euclidean metric).
The following construction defines a smooth function
that is positive on
and vanishes outside of
So in particular, if
is relatively compact then this function
will be a bump function.
If
then let
while if
then let
; so assume
is neither of these. Let
be an open cover of
by open balls where the open ball
has radius
and center
Then the map
defined by
is a smooth function that is positive on
and vanishes off of
For every
let
which is a real number because the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
vanishes at any
outside of
while on the compact set
the values of the partial derivatives are bounded.
[The partial derivatives are continuous functions so the image of the compact subset is a compact subset of The supremum is over all non-negative integers where because and are fixed, this supremum is taken over only finitely many partial derivatives, which is why ]
The series
converges uniformly on
to a smooth function
that is positive on
and vanishes off of
Moreover, for any non-negative integers
where this series also converges uniformly on
(because whenever
then the
th term's absolute value is
).
As a corollary, given two disjoint closed subsets
of
and smooth non-negative functions
such that for any
if and only if
and similarly,
if and only if
then the function
is smooth and for any
if and only if
if and only if
and
if and only if
In particular,
if and only if
so if in addition
is relatively compact in
(where
implies
) then
will be a smooth bump function with support in
Properties and uses
While bump functions are smooth, they cannot be
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
unless they
vanish identically . This is a simple consequence of the
identity theorem. Bump functions are often used as
mollifiers, as smooth
cutoff functions, and to form smooth
partitions of unity. They are the most common class of
test functions used in analysis. The space of bump functions is closed under many operations. For instance, the sum, product, or
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of two bump functions is again a bump function, and any
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
with smooth coefficients, when applied to a bump function, will produce another bump function.
If the boundaries of the Bump function domain is
, to fulfill the requirement of "smoothness" it have to preserve the continuity of all its derivatives, which leads to the following requirement at the boundaries of its domain:
The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a bump function is a (real) analytic function, and it can be extended to the whole complex plane: hence it cannot be compactly supported unless it is zero, since the only entire analytic bump function is the zero function (see
Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (18 ...
and
Liouville's theorem). Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of
for a large angular frequency
. The Fourier transform of the particular bump function
from above can be analyzed by a
saddle-point method
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point ( saddle point), in ...
, and decays asymptotically as
for large
.
[ Steven G. Johnson]
Saddle-point integration of ''C''∞ "bump" functions
arXiv:1508.04376 (2015).
See also
*
*
*
*
Citations
References
*
{{DEFAULTSORT:Bump Function
Smooth functions