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http://mathworld.wolfram.com/CompactSupport.html
Jan 02, 2020 · Compact Support. A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.
https://www.encyclopediaofmath.org/index.php/Function_of_compact_support
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 , …
https://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture14.pdf
The function f is compactly supported if supp fis compact. Notation. k C 0 (U) = The set of compactly supported Ck functions on U. (3.165) Suppose that f∈ Ck 0 ( U). Define a new set 1 = ( Rn−supp ). Then ∪ 1 = n, because supp f⊆ U. Define a new map f˜: Rn → R by f˜= f on U, (3.166) 0 on U 1. The function f˜is Ck on Uand Ck on U ˜ k 1, so fis in 0 (Rn).
http://www.ams.org/journals/tran/1971-156-00/S0002-9947-1971-0275367-4/S0002-9947-1971-0275367-4.pdf
Gillman and Jerison have shown that when A'is a realcompact space, the functions in C(X) with compact support are precisely the functions which belong to every free maximal ideal in C(X). This result, and other general background material, may be found in our basic reference [GJ].
https://math.stackexchange.com/questions/1344706/are-continuous-functions-with-compact-support-bounded
While studying measure theory I came across the following fact: K(X) ⊂ Cb(X) (meaning the continuous functions with compact support are a subset of the bounded continuous functions). This seems somehow odd to me; I've tried to prove it but did not succeed.
https://ncatlab.org/nlab/show/compact+support
Definition 0.1. A function on a topological space with values in a vector space (or really any pointed set with the basepoint called ) has compact support (or is compactly supported) if the closure of its support, the set of points where it is non-zero, is a compact subset. That is, the subset is a compact subset of . Typically,...
https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Start with a function ψ ∈ C∞c (Rd) having ψ = 1 on the ball B (0, 1), supported inside B (0, 2), and with 0 ≤ ψ ≤ 1 everywhere. Let M = maxi supB ( 0, 2) Diψ which is finite. Set ψn (x) = ψ (x / n). If f ∈ Y then clearly ψnf ∈ X for every n. Now ‖f − ψnf‖2H1 = ‖f − ψnf‖2L2 + ∑ i ‖Di (f − ψnf)‖2L2.
https://en.wikipedia.org/wiki/Support_function
The properties of the support function as a function of the set A are sometimes summarized in saying that : A h A maps the family of non-empty compact convex sets to the cone of all real-valued continuous functions on the sphere whose positive homogeneous extension is convex.
http://mathworld.wolfram.com/BumpFunction.html
A function that satisfies (1) and (2) is called a bump function. If then by rescaling, namely, one gets a sequence of smooth functions which converges to the delta function, providing that is a neighborhood of 0. SEE ALSO: Compact Support, Convolution, Delta Function, Smooth Function This entry contributed by …
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