  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.
