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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).
https://math.stackexchange.com/questions/2959431/the-continuous-functions-of-compact-support-are-dense-in-l1
approximation of 'any' bounded continuous function using bounded continuous functions with compact support Hot Network Questions Use Current Template as a Template Parameter to one of the Template Parameters
https://math.stackexchange.com/q/3518516
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https://mathproblems123.files.wordpress.com/2011/02/density-1.pdf
Oct 03, 2004 · Density of Continuous Functions in L1 October 3, 2004 1 Approximation by continuous functions In this supplement, we’ll show that continuous functions with compact support are dense in L1 = L1(Rn;m). The support of a complex valued function f on a metric space X is the closure of fx 2 X : f(x) 6= 0g.
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Chapter%2011-%20Convolutions%20and%20Approximations.pdf
such that µ(K) <∞when Kis a compact subset of X.Then Cc(X) (the space of continuous functions with compact support) is dense in Lp(µ) for all p∈[1,∞). Proof. First Proof. Let {Kk} ∞ k=1 be a sequence of compact sets as in Lemma 10.10 and set Xk= Ko k.Using Item 3. of Lemma 10.17, there exists {ψn,k} ∞ n=1 ⊂
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
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.
http://www.math.ucsd.edu/~bdriver/240A-C-03-04/Lecture_Notes/Older-Versions/chap22.pdf
22 Approximation Theorems and Convolutions 22.1 Density Theorems In this section, (X,M,µ) will be a measure space A will be a subalgebra of M. Notation 22.1 Suppose (X,M,µ) is a measure space and A ⊂M is a sub- ... of continuous functions with compact support) is dense in Lp ...
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 ...
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