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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?

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    @Alron: It would be great if you would write up what you learned and post it as an answer.2012-10-06

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We consider $1\leq p<\infty$. We can assume that $f\geq 0$ and $f$ is simple. We approximate the indicator function of a Borel subset $B$ of $\Bbb R^2$ by a finite union of product $\prod_{j=1}^NA_j\times B_j$ to get that $f$ and we approached in $L^p$ by functions of the form $\sum_{j=1}^Na_j\chi_{A_j\times B_j}$, where $a_j$ are real numbers and $A_j,B_j$, Borel sets of finite measure. The indicator of $A_j$ can be approched in $L^p$ by a test function $\phi_j(x)$, and the same for $B_j$.

So the set of functions of the form $\sum_{j=1}^Nf_j(x)g_j(y)$, where $f_j,g_j\in C_0^{\infty}(\Bbb R)$, $N\in\Bbb N$, is dense in $L^p(\Bbb R^2)$, with $1\leq p<\infty$.