Bump function

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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 graph of the bump function where and

Examples edit

 
The 1d bump function  

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

 
The non-analytic smooth function f(x) considered in the article.

Consider the function

 

defined for every real number x.


 
The smooth transition g from 0 to 1 defined here.

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).

A proper example of a smooth Bump function would be:

 

A proper example of a smooth transition function will be:

 

where could be noticed that it can be represented also through Hyperbolic functions:

 

Existence of bump functions edit

 
An illustration of the sets in the construction.

It is possible to construct bump functions "to specifications". Stated formally, if   is an arbitrary compact set in   dimensions and   is an open set containing   there exists a bump function   which is   on   and   outside of   Since   can be taken to be a very small neighborhood of   this amounts to being able to construct a function that is   on   and falls off rapidly to   outside of   while still being smooth.

Bump functions defined in terms of convolution

The construction proceeds as follows. One considers a compact neighborhood   of   contained in   so   The characteristic function   of   will be equal to   on   and   outside of   so in particular, it will be   on   and   outside of   This function is not smooth however. The key idea is to smooth   a bit, by taking the convolution of   with a mollifier. 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.

Bump functions defined in terms of a function   with support  

An alternative construction that does not involve convolution is now detailed. It begins by constructing a smooth function   that is positive on a given open subset   and vanishes off of  [1] This function's support is equal to the closure   of   in   so if   is compact, then   is a bump function.

Start with any smooth function   that vanishes on the negative reals and is positive on the positive reals (that is,   on   and   on   where continuity from the left necessitates  ); an example of such a function is   for   and   otherwise.[1] Fix an open subset   of   and denote the usual Euclidean norm by   (so   is endowed with the usual Euclidean metric). The following construction defines a smooth function   that is positive on   and vanishes outside of  [1] So in particular, if   is relatively compact then this function   will be a bump function.

If   then let   while if   then let  ; so assume   is neither of these. Let   be an open cover of   by open balls where the open ball   has radius   and center   Then the map   defined by   is a smooth function that is positive on   and vanishes off of  [1] For every   let

 
where this supremum is not equal to   (so   is a non-negative real number) because   the partial derivatives all vanish (equal  ) at any   outside of   while on the compact set   the values of each of the (finitely many) partial derivatives are (uniformly) bounded above by some non-negative real number.[note 1] The series
 
converges uniformly on   to a smooth function   that is positive on   and vanishes off of  [1] Moreover, for any non-negative integers  [1]
 
where this series also converges uniformly on   (because whenever   then the  th term's absolute value is  ). This completes the construction.

As a corollary, given two disjoint closed subsets   of   the above construction guarantees the existence of smooth non-negative functions   such that for any     if and only if   and similarly,   if and only if   then the function

 
is smooth and for any     if and only if     if and only if   and   if and only if  [1] In particular,   if and only if   so if in addition   is relatively compact in   (where   implies  ) then   will be a smooth bump function with support in  

Properties and uses edit

While bump functions are smooth, the identity theorem prohibits their 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   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:

 

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   for a large angular frequency  [2] The Fourier transform of the particular bump function

 
from above can be analyzed by a saddle-point method, and decays asymptotically as
 
for large  [3]

See also edit

Citations edit

  1. ^ The partial derivatives   are continuous functions so the image of the compact subset   is a compact subset of   The supremum is over all non-negative integers   where because   and   are fixed, this supremum is taken over only finitely many partial derivatives, which is why  
  1. ^ a b c d e f g Nestruev 2020, pp. 13–16.
  2. ^ 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.
  3. ^ Steven G. Johnson, Saddle-point integration of C "bump" functions, arXiv:1508.04376 (2015).

References edit