If $f(x_1,x_2)$ is in $L^p(\mathbb{R}^2)$, $p>1$, then can we find two $C_0^{\infty}$ functions $g(x_1)$ and $h(x_2)$ which defined in the real line that approximate $f$? If it is true, can we change the $L^p$ to the Sobolev space $W^{m,p}$ and still have this approximation?
density in $L^p$
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real-analysis
functional-analysis
sobolev-spaces
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0I guess you already know the result of density for $\Bbb R$. In this case, for $p<\infty$, we can assume $f\geq 0$ and simple. Then $f=\sum_{j=1}^Na_j\chi_{B_j}$. Approximate each $B_j$ by a finite sum of product of Borel sets, then the characteristic function of each member of the product, to get that $f$ can be approached by a finite sum of functions of the form $F(x,y)=f(x)g(y)$ with $f,g\in C_0^{\infty}$. – 2012-10-01
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2I don't understand the question. Surely, you cannot expect to approximate a function of two variables by a function of one variable? And what is the meaning of “definited”? – 2012-10-01
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0sorry, it should be "defined" – 2012-10-03
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0Could you explicitely write down the functions you want they approximate $f$? – 2012-10-03
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0I do not know what explicitly $f$, $g$ and $h$ are. But I get your idea. I think it works, and the answer is positive. Thank you. – 2012-10-04
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0@Alron: It would be great if you would write up what you learned and post it as an answer. – 2012-10-06