Simple fractions and linear decomposition of some convolutions of measures

Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form ${\displaystyle {\phi }_a(\xi )=\left[\frac{a}{h(\xi )+a}\right]}$, where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions of measures ${\displaystyle {\mu }_a}$ with ${\displaystyle \mu {̂}_a(\xi )={\phi }_a(\xi )}$ are linear combinations of powers of such measures. This can simplify calculations. It is interesting that this simplification uses signed measures since coefficients of linear combinations can be negative numbers. All the results of this paper except Proposition 1 remain true if we replace probability measures with complex valued measures with finite variation, and replace the characteristic function with Fourier transform.