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Concept
Bump function
Summary
In mathematics, a bump function (also called a test function) is a function on a Euclidean space which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain forms a vector space, denoted or The dual space of this space endowed with a suitable topology is the space of distributions.
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 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
Smooth transition functions
Consider the function
defined for every real number x.
The function
has a strictly positive denominator everywhere on the real line, hence g is also smooth. Furthermore, g(x) = 0 for x ≤ 0 and g(x) = 1 for x ≥ 1, hence it provides a smooth transition from the level 0 to the level 1 in the unit interval [0, 1]. 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.
Caution must be taken since, as example, taking , leads to:
which is not a infinitely differentiable function (so, is not "smooth"), so the constraints a < b < c < d must be strictly fulfilled.
Some interesting facts about the function:
Are that make smooth transition curves with "almost" constant slope edges (behaves like inclined straight lines on a non-zero measure interval).
Official source
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