analytic continuation of Fourier transform

Question

let be the Fourier transform

$$h(u)= \int_{-\infty}^{\infty} g(x) \exp(iux) \, \text{d}x$$

then let us suppose that for $ |x| \to \infty $ the function $g$ goes as

$$g(x) = \exp(-ax) \text{ for some positive $a$}$$

does it means that we can analytically continue the Fourier transform $h$ to the region of the complex plane where $-a < \text{Im}(z) < a$? Apart from being $h(u)$ defined for every real $u$?

if $g(x)$ tends to $0$ sufficiently fast does it mean I can define the fourier transform $h(z)$ for every complex number $z$?

in fact if we put $u = iz$ for real $z$ then

$$h(iz)= \int_{-\infty}^{\infty}g(x) \exp(-zx) \, \text{d}x$$ is it defined for every real $z$ in this case if $g$ is a Gaussian for example?

Answer 1

Yes, it is possible to continue $h$ analytically to $-a < \text{Im}(z) < a\ $ if $|g(x)| \le C\exp(-a|x|)\ $ for some $a>0$. The function is defined by putting $z$ instead of $x$ in the Fourier transform. If $g$ satisfy this estimate for any $a>0$ then $h$ is an entire function.