Question (1) Does the Fourier transform of a non-strictly positive real kernel
$f(t)$ always generate an entire function $g(z)$ with complex zeros?
$$g(z)=\int_{-\infty}^{\infty}f(t) \exp(izt)dt=2\int_{0}^{\infty}f(t) \cos(zt)$$
where $f(t)$ is continuous.
The function $f(t)$ behaves like the function $f_0(t)$ below:
$$f_0(t)=(1-(1/4)t^2)\exp(-t^2)$$
$f_0(t)$ has one real zero at $t_0=2$. So $f_0(t)<0$ when $t_0<t<\infty$ and $f_0(t) > 0$ when $0 < t < t_0$. $f(t) \to 0$ when $t \to \infty$.
