given any even function $ g(x)=g(-x) $ is it always possible to write it as the product of two functions ? i mean
$ g(x) = f(x)f(-x) $ so $ g(x) $ is always an even function even though $ f(x) $ it isn't
for example given the Riemann Xi function $ \xi(1/2+s)= \xi(1/2-s) $ can we represent it as the product of two functions ? $ \xi(1/2+s)=f(s)f(-s) $
in this case and using the Hadamard product $ f(x)= \prod _{n}(1- \frac{ix}{n}) $
On
There are many irreducible even functions over real numbers,which is a whole class of function that contradicts your statement.
On
As already mentioned by @pritam $g(0) < 0$ will lead to a contradiction. If we have $g(0) \ge 0$ the function
$$f(x)=\begin{cases}g(x) &; x > 0 \\ \sqrt{g(0)} &; x=0 \\ 1 &;x < 0 \end{cases}$$
would be a simple function $f(x)$ with $g(x)=f(x)f(-x)$.
If we allow complex values for $f$ then $f=\sqrt{g}$ would be a solution where $\sqrt{\cdot}$ is defined as one steam of the complex square root.
Consider $g(x)=-5+x^2$ then $g$ is even and $g(0)=-5<0$. Now if $g(x)=f(x)f(-x)$ for some $f$ then $g(0)=f(0)^2\geq 0$, a contradiction.