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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}) $

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    $g(0)=f(0)^2 \geq 0$ would be a necessary condition.2012-06-27

3 Answers 3