Bump Function
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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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
f: \R^n \to \R on a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\R^n which is both
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
(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 ...
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 In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalles ...
. 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 * Do ...
\R^n 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 \mathrm^\infty_0(\R^n) or \mathrm^\infty_\mathrm(\R^n). The dual space 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 \Psi:\R \to \R given by \Psi(x) = \begin \exp\left( -\frac\right), & x \in (-1,1) \\ 0, & \text \end 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 no ...
article. This function can be interpreted as the
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is ...
\exp\left(-y^2\right) scaled to fit into the unit disc: the substitution y^2 = / corresponds to sending x = \pm 1 to y = \infty. A simple example of a (square) bump function in n variables is obtained by taking the product of n copies of the above bump function in one variable, so \Phi(x_1, x_2, \dots, x_n) = \Psi(x_1) \Psi(x_2) \cdots \Psi(x_n).


Existence of bump functions

It is possible to construct bump functions "to specifications". Stated formally, if K is an arbitrary
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
in n dimensions and U 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 suf ...
containing K, there exists a bump function \phi which is 1 on K and 0 outside of U. Since U can be taken to be a very small neighborhood of K, this amounts to being able to construct a function that is 1 on K and falls off rapidly to 0 outside of K, while still being smooth. The construction proceeds as follows. One considers a compact neighborhood V of K contained in U, so K \subseteq V^\circ\subseteq V \subseteq U. 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 ...
\chi_V of V will be equal to 1 on V and 0 outside of V, so in particular, it will be 1 on K and 0 outside of U. This function is not smooth however. The key idea is to smooth \chi_V 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 \chi_V with a
mollifier In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
. The latter is just a bump function with a very small support and whose integral is 1. Such a mollifier can be obtained, for example, by taking the bump function \Phi from the previous section and performing appropriate scalings. An alternative construction that does not involve convolution is now detailed. Start with any smooth function c : \R \to \R that vanishes on the negative reals and is positive on the positive reals (that is, c = 0 on (-\infty, 0) and c > 0 on (0, \infty), where continuity from the left necessitates c(0) = 0); an example of such a function is c(x) := e^ for x > 0 and c(x) := 0 otherwise. Fix an open subset U of \R^n and denote the usual Euclidean norm by \, \cdot \, (so \R^n is endowed with the usual
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
). The following construction defines a smooth function f : \R^n \to \R that is positive on U and vanishes outside of U. So in particular, if U is relatively compact then this function f will be a bump function. If U = \R^n then let f := 1 while if U = \varnothing then let f := 0; so assume U is neither of these. Let \left(U_k\right)_^ be an open cover of U by open balls where the open ball U_k has radius r_k > 0 and center a_k \in U. Then the map f_k : \R^n \to \R defined by f_k(x) := c\left(r_k^2 - \left\, x - a_k\right\, ^2\right) is a smooth function that is positive on U_k and vanishes off of U_k. For every k \in \N, let M_k := \sup \left\, which is a real number because the supremum vanishes at any x outside of U_k while on the compact set \overline, the values of the partial derivatives are bounded.The partial derivatives \frac : \R^n \to \R are continuous functions so the image of the compact subset \overline is a compact subset of \R. The supremum is over all non-negative integers 0 \leq p = p_1 + \cdots + p_n \leq k where because k and n are fixed, this supremum is taken over only finitely many partial derivatives, which is why M_k < \infty. The series f := \sum_^ \frac converges uniformly on \R^n to a smooth function f : \R^n \to \R that is positive on U and vanishes off of U. Moreover, for any non-negative integers p_1, \ldots, p_n \in \Z, \frac f = \sum_^ \frac \frac where this series also converges uniformly on \R^n (because whenever k \geq p_1 + \cdots + p_n then the kth term's absolute value is \leq \frac = \frac). As a corollary, given two disjoint closed subsets A, B of \R^n and smooth non-negative functions f_A, f_B : \R^n \to [0, \infty) such that for any x \in \R^n, f_A(x) = 0 if and only if x \in A, and similarly, f_B(x) = 0 if and only if x \in B, then the function f := \frac : \R^n \to [0, 1] is smooth and for any x \in \R^n, f(x) = 0 if and only if x \in A, f(x) = 1 if and only if x \in B, and 0 < f(x) < 1 if and only if x \not\in A \cup B. In particular, f(x) \neq 0 if and only if x \in \R^n \smallsetminus A, so if in addition U := \R^n \smallsetminus A is relatively compact in \R^n (where A \cap B = \varnothing implies B \subseteq U) then f will be a smooth bump function with support in \overline.


Properties and uses

While bump functions are smooth, they cannot be analytic unless they vanish identically . This is a simple consequence of the
identity theorem In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...
. Bump functions are often used as
mollifier In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
s, as smooth
cutoff function In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) fun ...
s, and to form smooth
partitions of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
. They are the most common class of
test functions Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives d ...
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 with smooth coefficients, when applied to a bump function, will produce another bump function. If the boundaries of the Bump function domain is \partial x , 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: \lim_ \frac f(x) = 0,\,\forall n \geq 0, \,n \in \mathbb The Fourier transform 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 (189 ...
and Liouville's theorem). Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of 1/k for a large angular frequency , k, . The Fourier transform of the particular bump function \Psi(x) = e^ \mathbf_ 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 , k, ^ e^ for large , k, . Steven G. Johnson
Saddle-point integration of ''C'' "bump" functions
arXiv:1508.04376 (2015).


See also

* * * *


Citations


References

* {{DEFAULTSORT:Bump Function Smooth functions