[Math] On the Paley-Wiener theorem

Question

Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying the condition of the Paley-Wiener theorem that is even on the real line and whose restriction to $[0,\infty)$ is monotone?

Answer 1

The answer is yes :
Let $h$ be an even real valued Schwartz function whose Fourier transform has compact support. Then choose $f(y) = \int_{-\infty}^y x h(x)^{2} dx$ .