Bump function

which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.

forms a vector space, denoted

Note that the support of this function is the closed interval

, 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.

scaled to fit into the unit disc: the substitution

variables is obtained by taking the product of

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

The function has a strictly positive denominator everywhere on the real line, hence g is also smooth.

To have the smooth transition in the real interval [a, b] with a < b, consider the function For real numbers a < b < c < d, the smooth function equals 1 on the closed interval [b, c] and vanishes outside the open interval (a, d), hence it can serve as a bump function.

, leads to: which is not an infinitely differentiable function (so, is not "smooth"), so the constraints a < b < c < d must be strictly fulfilled.

make smooth transition curves with "almost" constant slope edges (a bump function with true straight slopes is portrayed this Another example).

Bump functions defined in terms of convolution The construction proceeds as follows.

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.

[1] This function's support is equal to the closure

and denote the usual Euclidean norm by

is endowed with the usual Euclidean metric).

the partial derivatives all vanish (equal

the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number.

As a corollary, given two disjoint closed subsets

the above construction guarantees the existence of smooth non-negative functions

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.

to fulfill the requirement of "smoothness", it has to preserve the continuity of all its derivatives, which leads to the following requirement at the boundaries of its domain:

Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of

from above can be analyzed by a saddle-point method, and decays asymptotically as