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:

$
\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.