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?



On the Paley-Wiener theorem
The answer is yes :  

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


