Determining if something is a characteristic function

Question

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$

Question: Given a function, say $\psi(t)$, how does one show that it is a characteristic function?

(Typed this on my phone - my apologies if there's poor formatting)

Answer 1

Let me just state the theorem I linked to in my comment, so that this question does not go unanswered.

Bochner's theorem

If $\varphi:\mathbb{R}^d\to \mathbb C$ is a complex-valued function with $\varphi(0)=1$, continuous at $0$ and nonnegative-definite in the sense that for $n\geq 1$ we have that $$ \sum_{j=1}^n\sum_{k=1}^n\varphi(z_j-z_k)\,\xi_j\bar{\xi}_k\geq 0,\quad \text{for }\;z_1,\ldots,z_n\in\mathbb{R}^d,\;\xi_1,\ldots,\xi_n\in\mathbb C, $$ then $\varphi$ is the characteristic function of a distribution (random variable) on $\mathbb{R}^d$.