On representation of the logarithm for arbitrary characteristic function on segments

We consider a characteristic function of arbitrary probability law. We obtain analogs of the Lévy–Khintchine formula for it on any segment of the form $[-r,r]$ with finite $r>0$, where the characteristic function does not vanish. Using these representations we prove a criterion of belonging of the corresponding distribution function to the new wide class of so called quasi-infinitely divisible distribution functions.