I want to decide for the following functions whether they are characteristic functions of a real-valued RV/probability measure on $\mathbb R$. $$\varphi(t) = \frac13e^{-8t^2}+\frac23e^{2(e^{3it}-1-it)}$$ and $$\psi(t) = \frac1{1+t^4}$$
I have already checked the basic properties $\varphi(-t)=\overline{\varphi(t)}$, $\varphi(0)=1$ and $\lvert\varphi\rvert\le 1$ for both functions, and they are both continuous.
I came across Bochner's theorem when looking for and approach to this problem. However, it is far beyond anything I have ever done and has never been introduced in any of my courses.
How can we (dis)prove that these functions are characteristic functions without Bochner's theorem?