A past question from a qual at my university reads: Prove that any function $f\in L^p([0,1]^2)$, $1\leq p<\infty$, can be approximated by a finite linear combination of functions of the form $h(x)g(x)$ with $h$ and $g$ continuous on $[0,1]$. More precisely, given $\epsilon>0$ there is a function
$$u(x,y)=\sum_{j=1}^n h_j(x)g_j(y)$$
with $h_j$ and $g_j$ continuous on $[0,1]$ for $j=1,2,\dots,n$, such that $||f-u||_p<\epsilon$.
I know that simple functions are dense in $L^p$. Further, I know that because $|f|^p\in L^1([0,2]^2)$, then for any $\epsilon>0$, there is a continuous function $g$, that vanishes outside of a bounded set such that $\int_{[0,1]^2}||f|^p-g|<\epsilon$. However, I don't know how to get the approximating function to be in the form specified in the question. How should I proceed?
Best Answer
You should replace $h(x)g(x)$ by $h(x)g(y)$. Otherwise the result is false.
First approximate $f$ by a continuous function $\phi$. Then use Stone -Weierstrass Theorem. Linear combinations of functions of the form $h(x)g(y)$ form an algebra of continuous functions which separates points and contains constants. Hence $\phi$ can be approximated uniformly by linear combinations of functions of the form $h(x)g(y)$. Finally approximation in the uniform metric implies approximation in $L^{p}$.