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Function of compact support

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A function defined in some domain of , having compact support belonging to this domain. More precisely, suppose that the function is defined on a domain . The support of is the closure of the set of points for which is different from zero . Thus one can also say that a function of compact support in is a function defined on such that its support is a closed bounded set located at a distance from the boundary of by a number greater than , where is sufficiently small.

One usually considers -times continuously-differentiable functions of compact support, where is a given natural number. Even more often one considers infinitely-differentiable functions of compact support. The function

can serve as an example of an infinitely-differentiable function of compact support in a domain containing the sphere .

The set of all infinitely-differentiable functions of compact support in a domain is denoted by . On one can define linear functionals (generalized functions, cf. Generalized function). With the aid of functions 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 is dense in some concrete space of functions. It is known, for example, that if the boundary of a bounded domain is sufficiently smooth, then is dense in the space of functions

(), that is, in the Sobolev space of functions of class that vanish on along with their normal derivatives of order up to and including ().

References

[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)


Comments

References

[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. S.M. Nikol'skii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Function_of_compact_support&oldid=11539
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098