Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable?
This is becasue I want to apply a convolution whose kernel is define that way, and I am willing to separate that kernel. For that I need to be able to express it as a product of two functions that are on each variable separatedly.
It cannot be done. Suppose to the contrary that it can be done. We will derive a contradiction. Suppose that $\frac{x^2}{\sqrt{x^2+y^2}}=f(x)g(y)$ for some functions $f$ and $g$. Then $$f(1)g(1)=\frac{1}{\sqrt{2}},$$ and $$f(1)g(2)=\frac{1}{\sqrt{5}}$$ and $$f(2)g(1)=\frac{4}{\sqrt{5}}$$ Note $$f(2)g(2)=\frac{f(1)g(2)f(2)g(1)}{f(1)g(1)}=\sqrt{2}\cdot \frac{4}{5}.$$ But $f(2)g(2)=\frac{2^2}{\sqrt{2^2+2^2}}=\frac{1}{\sqrt{2}}$.