Editing Bump function (section)

==Properties and uses==

While bump functions are smooth, the [[identity theorem]] prohibits their being [[analytic function|analytic]] unless they [[Zero of a function|vanish]] identically. Bump functions are often used as [[mollifier]]s, as smooth [[cutoff function]]s, 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.

If the boundaries of the Bump function domain is <math>\partial x,</math> 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:
<math display="block">\lim_{x \to \partial x^\pm} \frac{d^n}{dx^n} f(x) = 0,\,\text { for all } n \geq 0, \,n \in \Z</math>

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 (complex analysis)|Liouville's theorem]]).  Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of <math>1/k</math> for a large angular frequency <math>|k|.</math><ref>K. O. Mead and L. M. Delves, "On the convergence rate of generalized Fourier expansions," ''IMA J. Appl. Math.'', vol. 12, pp. 247–259 (1973) {{doi|10.1093/imamat/12.3.247}}.</ref>  The Fourier transform of the particular bump function
<math display="block">\Psi(x) = e^{-1/(1-x^2)} \mathbf{1}_{\{|x|<1\}}</math>
from above can be analyzed by a [[saddle-point method]], and decays asymptotically as
<math display="block">|k|^{-3/4} e^{-\sqrt{|k|}}</math>
for large <math>|k|.</math><ref>[[Steven G. Johnson]], [https://arxiv.org/abs/1508.04376 Saddle-point integration of ''C''<sub>∞</sub> "bump" functions], arXiv:1508.04376 (2015).</ref>