Consider the integral $$F(x) = \int_0^1 f(k) \exp(i g(k) x) dk$$ where $i$ is the imaginary unit, and $g(k)$ is a real function that may have poles in the integration interval.
I suspect the following is true, but can't prove it
$F(x)$ exists for all $x$ $\implies$ $f(k)$ has a zero wherever $g(k)$ has a pole
Is it true?