What functions can be characteristic functions of real-valued random variables? For a real-valued random variable $X$, the characteristic function $\phi \colon \mathbb{R} \to \mathbb{C}$ is defined as
$$\phi(t) = \mathbb{E}[e^{itX}].$$
One can check that $\phi$ satisfies


*

*$\phi(0) = 1$,

*$\phi$ is uniformly continuous,

*$\phi(t) \to 0$ as $|t| \to \infty$.


Conversely, if we have a $\phi$ satisfying these three properties, is there a distribution with characteristic functions $\phi$? In my specific case, I have a $\phi \in C^\infty \cap L^1 \cap L^\infty$.
 A: You want Bochner's theorem for necessary and sufficient conditions.  The conditions you give do not ensure non-negativity of the measure with characteristic function $\phi$.  Bochner's theorem does, by requiring also that the matrices $(\phi(t_i-t_j))$ be positive semi-definite.  The necessity is easy to see.  Consider expectations of form $$E\left|\sum_{k=0}^na_k \exp(i t_k X)\right|^2 = \sum_{k=0}^n\sum_{l=0}^n E (\exp(i(t_k-t_l)X) a_k \overline a_l ) =\sum_{k=0}^n\sum_{l=0}^n \phi(t_k-t_l)  a_k \overline a_l,$$ for arbitrary $a_k\in\mathbb C$. Such sums must be non-negative for all choices of the $a_k$, and hence $\phi$ must be positive definite. 
By the way, the limit $\phi(t)\to0$ as $|t|\to\infty$ that you state, does not hold for all real random variables.  If $P(X=1)=P(X=-1)=1/2$, then $\phi(t)=\cos(t)$, for example. Or even more simply: if $P(X=0)=1$ then $\phi(t)=1$ for all $t$.
In your case, where $\phi\in C^\infty\cap L^1 \cap L^\infty$, there is a function $f$, not necessarily non-negative, such that $\phi(t)=\int \exp(itx)f(x)\,dx$.  But without checking something equivalent to Bochner's condition that the kernel $K(s,t)=\phi(s-t)$ is positive definite, you won't know that $f$ is non-negative.
For example, the function $\phi(t)=1/(1+t^2)$ satisfies your conditions, and is the characteristic function of a probability distribution.  But the function $\psi(t)=2\phi(t)-\phi(2t)$ also satisfies the conditions and is not the characteristic function of any probability distribution.  
