Random variable $\xi$ is distributed by symmetrical principle with density function $\frac{1}{2a} \mathcal{1}_{[-a,a]}(x),$ here $a>0$. I need to find characteristic function.
I never seen anything like this before. I tried to find some theory about it. And I found that $f(t)=\int_{-\infty}^{+\infty}e^{itx} dF(X)$, where $f(t)$ is characteristic function and $F(x)$ is distribution function. Also I found that density function is $p(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{-itx} f(t)dt.$ Do I can do something with this? How could I find characteristic function?
For continuous distributions $dF=pdx$ on the support, in this case $[-a,\,a]$, so $$f(t)=\int_{-a}^a\frac{1}{2a}\exp itx dx=\left[\frac{\exp itx}{2ita}\right]_{-a}^a=\frac{1}{ta}\sin ta$$(or $1$ at $t=0$ by continuity).