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https://math.stackexchange.com/questions/67370/smooth-functions-with-compact-support-are-dense-in-l1
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https://en.wikipedia.org/wiki/Smooth_function
Smooth functions with given closed support are used in the construction of smooth partitions of unity (see partition of unity and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence.
https://en.wikipedia.org/wiki/Bump_function
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 and closed support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article.
https://www.planetmath.org/SmoothFunctionsWithCompactSupport
Then the set of smooth functions with compact support (in U) is the set of functions f: ℝ n → ℂ which are smooth (i.e., ∂ α f: ℝ n → ℂ is a continuous function for all multi-indices α) and supp f is compact and contained in U. This function space is denoted by C 0 ∞ (U).
https://math.rice.edu/~semmes/fun5.pdf
Some basic aspects of smooth functions and distributions on open subsets of Rn are briefly discussed. Contents 1 Smooth functions 2 2 Supremum seminorms 3 3 Countably many seminorms 4 4 Cauchy sequences 5 5 Compact support 6 6 Inductive limits 8 7 Distributions 9 8 Differentiation of distributions 10 9 Multiplication by smooth functions 11
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://mathoverflow.net/questions/224940/spectrum-of-ring-of-smooth-functions-on-mathbbrn/224941
Together with the obvious fact that smooth functions are continuous, this is enough to see that the embedding $\psi$ is indeed a homeomorphism onto its image. Note that $\text{Spec}(R)$, or even, for that matter, the maximal ideals $\text{mSpec}(R)$, are much bigger than the image of $\psi$.
http://texas.math.ttu.edu/~gilliam/f06/m5340_f06/mollifiers_approx.pdf
Mollifiers and Approximation by Smooth Functions with Compact Support Let ρ∈ C∞(Rn) be a non-negative function with support in the unit ball in Rn.In particular we assume that ρ(x) ≥ 0 for x∈ Rn, ρ(x) = 0 for kxk >1, and Z Rn ρ(x)dx= 1.
https://www.chebfun.org/examples/approx/SmoothCompact.html
Smooth functions of compact support Nick Trefethen, July 2014 in approx download ...
https://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture14.pdf
Lecture 14 As before, let f: R R→ be the map defined by 0 f(x) = e−1/x if x≤ 0, ... 3.9 Support and Compact Support Now for some terminology. Let U be an open set in Rn, and let f : U → R be a continuous function. Definition 3.26. The support of fis ... There exist functions f
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