Turn aperiodic function to periodic I got a quick question regarding aperiodic functions. Let's suppose I have an aperiodic function 
$$
f(x) = \left\{
  \begin{array}{l l}
    \exp(-t) & \quad -2\leq t \leq2,\\
    0 & \quad \text{anywhere else,}
  \end{array} \right.
$$
and I want this to become periodic with period $\ T=4 $. What are the steps involved in such a conversion? I know that $\ f(x+nT) = f(x) $ must hold in resulting function but I am really stuck at the moment. Thank you in advance!

You don't really "convert" it to a periodic function.  You produce a periodic function that agrees with $f$ on a certain interval.  Perhaps that interval
should be $[-2,2)$.  So if $g$ is the periodic function, you want $g(t + 4 n) = f(t) = \exp(-t)$ for $-2 \le t < 2$ and integer $n$.  You could take 
$g(x) = f(h(x))$ where $h$ is a "sawtooth" function, $h(x) = t$ where $-2 \le t < 2$ and $(x-t)/4$ is an integer.  Except at the discontinuities, you could take
$h(x) = \dfrac{\pi}{4} \arctan(\tan(\pi x/4))$.
