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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 , where is sufficiently small.
https://www.planetmath.org/SmoothFunctionsWithCompactSupport
Then the set of smooth functions with compact support (in U) is the set of functions f:ℝn→ℂ which are smooth (i.e., ∂αf:ℝn→ℂ is a continuous function for all multi-indices α) and suppf is compact and contained in U. This function space is denoted by C0∞(U).
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).
https://www.sciencedirect.com/topics/mathematics/function-with-compact-support
For smooth functions with compact support (2.0.1) can be shown by integration by parts. The general case is treated by approximation. A first proof was given by Korn in. We note that variants of Korn's inequality in L 2 have been established by Courant and Hilbert, Friedrichs, Èidus and Mihlin.
https://math.stackexchange.com/questions/284045/example-of-a-function-with-compact-support
example of a function with compact support. Ask Question Asked 6 years, 11 months ago. Active 6 years, 11 months ago. Viewed 3k times 0. 3 $\begingroup$ ... The Fourier transform of functions with compact support is differentiable. 1. Do the hat functions have compact support? 3.
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://www.encyclopediaofmath.org/index.php/Support_of_a_function
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
https://www.sciencedirect.com/science/article/pii/S0079816908602779
The chapter mentions that the Fourier transform of a continuous function with compact support can be extended to the complex space C n, as an entire analytic function of exponential type. The chapter also describes the analogy between Paley-Wiener theorem, and the theorem on the Fourier-Borel transformation of the analytic functionals.
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