A function defined in some domain of $ E ^ $, having compact support belonging to this domain. More precisely, suppose that the function $ f ( x) = f ( x _ \dots x _ ) $ is defined on a domain $ \Omega \subset E ^ $. The support of $ f $ is the closure of the set of points $ x \in \Omega $ for which $ f ( x) $ is different from zero $ ( f ( x) \neq 0) $. Thus one can also say that a function of compact support in $ \Omega $ is a function defined on $ \Omega $ such that its support $ \Lambda $ is a closed bounded set located at a distance from the boundary $ \Gamma $ of $ \Omega $ by a number greater than $ \delta > 0 $, where $ \delta $ is sufficiently small.
One usually considers $ k $- times continuously-differentiable functions of compact support, where $ k $ is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function
$$ \psi ( x) = \ \left \< \begin e ^ - 1) > , & | x | < 1, | x | ^ = \sum _ ^ < n >x _ ^ , \\ 0 , & | x | \geq 1 , \\ \end \right .$$
can serve as an example of an infinitely-differentiable function of compact support in a domain $ \Omega $ containing the sphere $ | x | \leq 1 $.
The set of all infinitely-differentiable functions of compact support in a domain $ \Omega \subset E ^ $ is denoted by $ D $. On $ D $ one can define linear functionals (generalized functions, cf. Generalized function). With the aid of functions $ v \in D $ one can define generalized solutions (cf. Generalized solution) of boundary value problems.
In theorems concerned with problems on finding generalized solutions, it is often important to know whether $ D $ is dense in some concrete space of functions. It is known, for example, that if the boundary $ \Gamma $ of a bounded domain $ \Omega \subset E ^ $ is sufficiently smooth, then $ D $ is dense in the space of functions
$$ > <> _
^ ( \Omega ) = \ \left \ < : < f \in W _
^ ( \Omega ),\ \left . \frac <\partial ^ f > <\partial n ^ > \right | _ \Gamma = 0,\ s = 0 \dots r - 1 > \right \> $$
( $ 1 \leq p \leq \infty $), that is, in the Sobolev space of functions of class $ W _
^ ( \Omega ) $ that vanish on $ \Gamma $ along with their normal derivatives of order up to and including $ r - 1 $( $ r = 1, 2 ,\dots $).
| [1] | V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian) |
| [2] | S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) |
| [a1] | L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951) |
| [a2] | J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French) |
| [a3] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 |
How to Cite This Entry:
Function of compact support. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_compact_support&oldid=47009
This article was adapted from an original article by S.M. Nikol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article