On Zeros of Fourier Transforms

In this work we verify the sufficiency of a Jensen's necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast decreasing functions. We also apply our result to several important special functions in mathematics, such as modified Bessel function $K_{iz}(a),\ a>0$ as a function of variable $z$, Riemann Xi function $\Xi(z)$, and character Xi function $\Xi(z;\chi)$ when $\chi$ is a real primitive non-principal character satisfying $\varphi(u;\chi)\ge0$ on the real line, we prove these entire functions have only real zeros.