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Suppose I have an integrable non-const function $f(x,y)$ defined on $V$. I'd like to approximate it with a product $a(x)b(y)$. How can I find such $a(x)$ and $b(y)$ that $$\tag1 \iint_V\left(f(x,y)-a(x)b(y)\right)^2\,dxdy=min$$ ? Are $a$ and $b$ unique for this problem (not distinguishing $a$ and $b$ from $ca$ and $\frac1c b$)?

I've tried taking average of $f$ in $y$ and $x$ directions and use these averages as $a$ and $b$, but I can't really say if they satisfy $(1)$, in fact I'm sure they are much worse than one could find.

Ruslan
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  • Uniqueness should be understood up to a constant, because you can also take $2a$ and $(1/2)b$. – 40 votes Jul 14 '13 at 20:41
  • If $f$ is fixed and you seek $a,b$ maybe this is related to calculus of variations, since the expression is some kind of functional depending on $a,b$. Of course it's a functional of two variable functions, in some sense. – coffeemath Jul 14 '13 at 22:57

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