two functions $ f(x) $ and $ g(x) $

Question

let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $

also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ g(s)=g(1-s) $

then we use some numerical method to evaluate the roots and we check that the first 100 roots agree so can we conclude that $ f(x)= g(x) $ or at least $ f(x)=h(x)g(x) $ they are proportional function

Answer 1

You can almost never conclude that some equation is true by just checking with numerical methods. For instance if:

$$ f = \mathbb 1_{\mathbb R \backslash {\mathbb Q}} $$

Any computer will give you $f = 0$.

Moreover, in your case, $g = \mathbb 1_{{\mathbb Q}}$ gives a counter-example.