Is it possible to construct a characteristic function (for a distribution) $\phi(t)$ such that $\phi(t) = t^{-1/4}$ for $16\leq t \leq 20$?
Pólya’s theorem can be used to construct such a $\phi$. Here's a simple example:
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$ \phi(t) = \begin{cases} 1 - \frac{t}{32} & \text{if } 0 \leq t < 16 \\ \\ t^{-\frac{1}{4}} & \text{if } t \geq 16 \\ \\ \phi(-t) & \text{if t} \lt 0 \end{cases} $
It's easily verified that the conditions of the theorem are met, so this $\phi$ is the characteristic function of an absolutely continuous symmetrical distribution.