Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below:
$\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$
Can it be proven that the result is also a characteristic function? If so, I am guessing this result could extend to any number of combinations $\alpha_i$ as long as $\sum_i\alpha_i=1$
Secondly, if $\phi$ is again a characteristic function, then $\mathfrak{R}e\phi(t)=\frac12(\phi(t)+\phi(-t))$ is also a characteristic function. I don't even know how to begin attempting this proof as I am not sure what the $\mathfrak{R}$ represents.
Lastly, regarding the symmetry of characteristic functions,
$\phi$ is symmetric about zero iff it is real-valued iff the corresponding distribution is symmetric about zero.
Once again, my lack of familiarity with the complex plane leaves me in the dark here. Why can a complex-valued function not be symmetric about zero?