I am looking for explicit families of entire functions of exponential type A which are $L^2$ on the real axis. If not families, at least specific explicit examples, or guidelines about how to determine explicitly such functions.
More specifically, I am looking for a entire function of exponential type A which is $L^2$ on the real axis, whose restriction on the real axis is real-valued, even, bell-shaped and strictly positive (it can oscillate though).
Any kind of help is very welcome (please if you know references which give explicit examples of such functions or similar functions, include them in your answer)!