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### Smooth functions with compact support are dense in $L^1$

https://math.stackexchange.com/questions/67370/smooth-functions-with-compact-support-are-dense-in-l1
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### Are compactly supported continuous functions dense in the ...

https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Continuous functions on $\mathbb R^d$ such that the support is a compact subset of $\overline{\Omega}$? For "nice" $\Omega$ this would be the space of continuous functions on $\Omega$ vanishing at the boundary. $\endgroup$ – Jochen Wengenroth Apr 29 '16 at 12:50

### compactly supported continuous functions are dense in L^p

https://www.planetmath.org/CompactlySupportedContinuousFunctionsAreDenseInLp
Now, it follows easily that any simple function ∑ i = 1 n c i ⁢ χ A i, where each A i has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in L p ⁢ (X) we see that C c ⁢ (X) is also dense in L p ⁢ (X).

### Function of compact support - Encyclopedia of Mathematics

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 , …

### $C^\infty$ dense in Sobolev spaces - Mathematics Stack ...

https://math.stackexchange.com/questions/184129/c-infty-dense-in-sobolev-spaces
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### Support (mathematics) - Wikipedia

https://en.wikipedia.org/wiki/Support_(mathematics)
In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero. If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis

### Lp 4. Dense Subspaces of Lp L E Lp n

http://www.math.nthu.edu.tw/~kchen/teaching/5131week3.pdf
dense in Lp(E). These step functions are linear combinations of characteristic functions on some dyadic cubes. This implies that the space of simple functions is also dense in Lp(Rn). In this section we prove that the space of smooth functions with compact supports, and the space of functions with rapidly decreasing derivatives are also dense ...

### Molliﬁers and Approximation by Smooth Functions with ...

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.

### Smoothness - Wikipedia

https://en.wikipedia.org/wiki/Smooth_function
A bump function is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere (hence continuous).

### Distribution (mathematics) - Wikipedia

https://en.wikipedia.org/wiki/Distribution_(mathematics)
The basic space of test function consists of smooth functions with compact support, leading to standard distributions. Use of the space of smooth, rapidly (faster than any polynomial increases) decreasing test functions (these functions are called Schwartz functions ) gives instead the tempered distributions, which are important because they ...

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