Bump function

An illustration of a bump function in two variables.

In mathematics, a bump function is a function $f:{\mathbb {R}}^{n}\to {\mathbb {R}}$ on a Euclidean space ${\mathbb {R}}^{n}$ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The space of all bump functions on ${\mathbb {R}}^{n}$ is denoted $C_{0}^{\infty }({\mathbb {R} }^{n})$. The dual space of this space is the space of distributions.

Examples

The function Ψ(x).

The function $\Psi :\mathbb {R} \to \mathbb {R}$ given by

$\Psi (x)={\begin{cases}e^{-1/(1-x^{2})}&{\mbox{ for }}|x|<1\\0&{\mbox{ otherwise}}\end{cases}}$

is an example of a bump function in one dimension. It is clear from the construction that this function has compact support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article.

A simple example of a 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

An illustration of the sets in the construction.

It is possible to construct bump functions "to specifications". Stated formally, if $K$ is an arbitrary compact set in n dimensions and $U$ is an open set 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 $K$ that contains $U,$ so $K\subset V^{o}\subset V\subset U.$ The characteristic function $\chi _{V}$ 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 of $\chi _{V}$ with a mollifier. 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.

Properties and uses

While bump functions are smooth, they cannot be analytic unless they vanish identically. This is a simple consequence of analytic continuation.

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 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. However, the Fourier transform of a bump function will in general only be a Schwartz function rather than a bump function.