Does inverse wrapping exist? function $f$ on $R$ can be made periodic with period 1 by wrapping 
$$g(x)=\sum_{k=-\infty}^{\infty}f(x+k)$$ 
Given $g$ with period 1. How to find continuous $f$ in general case? 
 A: The solution to your problem is non-unique. But if you just want a candidate:
assuming $g$ is continuous, let $c$ be the average value of $g$ over $[0,1]$. Then by intermediate value theorem, the function $g - c$ hits 0 at some point. By translating the axis we can assume that $g-c = 0$ at $0$. Define $f_1(x) = g-c$ if $x\in [0,1]$ and $0$ elsewhere. $f_1$ is continuous, has compact support, and its periodicization is $g-c$. 
Now if suffices to find a continuous function $f_2$ such that its periodicization is $c$. For this we simply define 
$$ f_2 = \begin{cases}
cx & x\in[0,1]\\
c - c(x-1) & x\in(1,2]\\
0 & x\notin [0,2]
\end{cases} $$
You easily check that $f_2$ is continuous, has compact support, and has periodicization the constant value $c$. So $f = f_1 + f_2$ will "wrap" to $g$.
(By suitable mollification, $f_2$ can clearly be made smooth. By considering derivatives, you can make $f_1$ as smooth as the function $g$.)

Edit:
After thinking about this more, a slightly simpler construction is as follows. Observe that we assumed $g$ to be a continuous, periodic function. Extend $g$ to over $\mathbb{R}$ by periodicity. Then it suffices to construct a function $\psi$, with the following property:


*

*$\sum_{k = -\infty}^\infty \psi(x+k) = 1$

*$\psi$ is continuous (smooth)


This is because if you let $f(x) = g(x) \psi(x)$, using periodicity of $g$ and the above property of $\psi$, $f$ has the desired properties. Now let $\eta$ be a smooth transition function: $\eta = 0$ if $x < 0$, $\eta = 1$ if $x > 1$, and $\eta$ is smooth. Consider 
$$ \phi(x) = \frac{1}c \eta(x) (1- \eta(x - k)) $$
where $k$ is a positive integer and the constant $c$ is chosen such that the compactly supported function $\phi(x)$ verifies the condition above on its sum. (By construction the periodic sum of $\eta(x)(1- \eta(x-k))$ is some constant integer.)  
