Bump function

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.

. The transition function g above can also be written explicitly as w(x)=\begin{cases}\frac{1}{1+e^{\frac{2x-1}{x^2-x}}}&\text{if }0 and, on 0 , its nonconstant branch can be represented using Hyperbolic functions: \frac{1}{1+e^{\frac{2x-1}{x^2-x}}} = \frac{1}{2}\left( 1-\tanh\left(\frac{2x-1}{2(x^2-x)} \right) \right) Notice that u(x) could be used to built extended flat-top smooth bump functions, as example, for an integer value M>0 : y(x) = \sum_{k=-M}^{M} u(x+k) will have an unitary flat-top on x \in [-M,\,M] while being zero on |x|\geq M+1 , somehow working similar to a Partition of unity. ==Existence of bump functions== It is possible to construct bump functions "to specifications". Stated formally, if K\subset\mathbb{R}^n is compact and U\subset\mathbb{R}^n 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. Bump functions defined in terms of convolution The construction proceeds as follows. One considers a compact neighborhood V of K contained in U, so K \subseteq V^\circ\subseteq V \subseteq 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. Bump functions defined in terms of a function c : \Reals \to [0, \infty) with support (-\infty, 0] An alternative construction that does not involve convolution is now detailed. It begins by constructing a smooth function f : \Reals^n \to \Reals that is positive on a given open subset U \subseteq \Reals^n and vanishes off of U. This function's support is equal to the closure \overline{U} of U in \Reals^n, so if \overline{U} is compact, then f is a bump function. Start with any smooth function c : \Reals \to \Reals that vanishes on the negative reals and is positive on the positive reals (that is, c = 0 on (-\infty, 0) and c > 0 on (0, \infty), where continuity from the left necessitates c(0) = 0 ); an example of such a function is c(x) := e^{-1/x} for x > 0 and c(x) := 0 otherwise. Fix an open subset U of \Reals^n and denote the usual Euclidean norm by \|\cdot\| (so \Reals^n is endowed with the usual Euclidean metric). The following construction defines a smooth function f : \Reals^n \to \Reals that is positive on U and vanishes outside of U. So in particular, if U is relatively compact then this function f will be a bump function. If U = \Reals^n then let f = 1 while if U = \varnothing then let f = 0 ; so assume U is neither of these. Let \left(U_k\right)_{k=1}^\infty be an open cover of U by open balls where the open ball U_k has radius r_k > 0 and center a_k \in U. Then the map f_k : \Reals^n \to \Reals defined by f_k(x) = c\left(r_k^2 - \left\|x - a_k\right\|^2\right) is a smooth function that is positive on U_k and vanishes off of U_k. For every k \in \mathbb{N}, let M_k = \sup \left\{\left|\frac{\partial^p f_k}{\partial^{p_1} x_1 \cdots \partial^{p_n} x_n}(x)\right| ~:~ x \in \Reals^n \text{ and } p_1, \ldots, p_n \in \Z \text{ satisfy } 0 \leq p_i \leq k \text{ and } p = \sum_i p_i\right\}, where this supremum is not equal to +\infty (so M_k is a non-negative real number) because \left(\Reals^n \setminus U_k\right) \cup \overline{U_k} = \Reals^n, the partial derivatives all vanish (equal 0 ) at any x outside of U_k, while on the compact set \overline{U_k}, the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number. The series f ~:=~ \sum_{k=1}^{\infty} \frac{f_k}{2^k M_k} converges uniformly on \Reals^n to a smooth function f : \Reals^n \to \Reals that is positive on U and vanishes off of U. Moreover, for any non-negative integers p_1, \ldots, p_n \in \Z, \frac{\partial^{p_1+\cdots+p_n}}{\partial^{p_1} x_1 \cdots \partial^{p_n} x_n} f ~=~ \sum_{k=1}^{\infty} \frac{1}{2^k M_k} \frac{\partial^{p_1+\cdots+p_n} f_k}{\partial^{p_1} x_1 \cdots \partial^{p_n} x_n} where this series also converges uniformly on \Reals^n (because whenever k \geq p_1 + \cdots + p_n then the k th term's absolute value is \leq \tfrac{M_k}{2^k M_k} = \tfrac{1}{2^k} ). This completes the construction. As a corollary, given two disjoint closed subsets A, B of \Reals^n, the above construction guarantees the existence of smooth non-negative functions f_A, f_B : \Reals^n \to [0, \infty) such that for any x \in \Reals^n, f_A(x) = 0 if and only if x \in A, and similarly, f_B(x) = 0 if and only if x \in B, then the function h ~:=~ \frac{f_A}{f_A + f_B} : \Reals^n \to [0, 1] is smooth and for any x \in \Reals^n, h(x) = 0 if and only if x \in A, h(x) = 1 if and only if x \in B, and 0 if and only if x \not\in A \cup B. In particular, h(x) \neq 0 if and only if x \in \Reals^n \smallsetminus A, so if in addition U := \Reals^n \smallsetminus A is relatively compact in \Reals^n (where A \cap B = \varnothing implies B \subseteq U ) then h will be a smooth bump function with support in \overline{U}. ==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. If the boundaries of the Bump function domain is \partial x, 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: \lim_{x \to \partial x^\pm} \frac{d^n}{dx^n} f(x) = 0,\,\text { for all } n \geq 0, \,n \in \Z 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). Because the bump function is infinitely differentiable, its Fourier transform must decay faster than any finite power of 1/k for a large angular frequency |k|. The Fourier transform of the particular bump function \Psi(x) = e^{-1/(1-x^2)} \mathbf{1}_{\ } for large |k|. The integral of the bump function \Psi(x) is given by \int_{-\infty}^{\infty} \Psi(x) d x = e^{-1/2} \left( K_1\left(\frac{1}{2}\right) - K_0\left(\frac{1}{2}\right)\right) where K_1(x) and K_0(x) are the Modified Bessel functions of the second kind. ==See also== ==Citations== ==References==

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