In mathematical analysis, a bump function is a localized auxiliary function, usually chosen to be smooth and to have compact support. Bump functions are commonly used as cutoff functions, for example functions that are equal to 1 on a prescribed set and vanish outside a larger set, and as standard examples of kernels used to construct mollifiers.

Some authors use the term more broadly for any compactly supported smooth function. Such functions are important examples of test functions, especially in distribution theory, but the terms "bump function" and "test function" are not synonymous in all contexts.

Examples

. In fact, by definition of support, where the closure is taken with respect the Euclidean topology of the real line. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function

. This function is supported on the unit ball centered at the origin.

and zero elsewhere, for example

1. .

Smooth transition functions

A standard starting point is the function

defined for every real number x.

From this, define

The denominator is strictly positive everywhere on the real line, so g is smooth. Moreover, g(x)=0 for x≤0 and g(x)=1 for x≥1, so g gives a smooth transition from 0 to 1 on the unit intervala, b with a<b:

A compactly supported bump function can be obtained by multiplying a rising transition by a falling transition. For real numbers , the function

b, c, and vanishes outside the open interval (a, d). Thus it can serve as a bump function. When the plateau has positive length; when it degenerates to a single point, but the function is still smooth.

gives the smooth bump function

The formula

.

A related parameterized interior expression is

1. .

, with suitable endpoint values supplied, gives smooth transition curves with "almost" constant slope edges. A bump function with true straight slopes is portrayed by this example.

The transition function g above can also be written explicitly as

, its nonconstant branch can be represented using Hyperbolic functions:

Existence of bump functions

is compact is an open set while still being smooth.

Bump functions defined in terms of convolution

The characteristic function a bit, by taking the convolution with a mollifier from the previous section and performing appropriate scalings.

is a bump function.

otherwise. and denote the usual Euclidean norm is endowed with the usual Euclidean metric will be a bump function.

be an open cover where this supremum the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number.th term's absolute value). This completes the construction.

Properties and uses

While bump functions are smooth, the identity theorem prohibits them from being analytic unless they vanish identically. 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.

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 and Liouville's theorem from above can be analyzed by a saddle-point method

are the Modified Bessel functions of the second kind.

See also

Citations

References