Construction of Mikusinski operational calculus based on the convolution algebra of distributions. The theorems and the beginning of use

We consider the Mikusinski operational calculus based on the convolution algebra of distributions $D^\prime_+$ and $D^\prime_-$. We state and prove the basic theorems, and give examples of Mikusinski operational calculus using, which demonstrate its additional possibilities, such as extension of solutions to the domain of negative argument values, removing the growth limits of right-hand functions and obtaining the new methods for solving the nonhomogeneous equations with discontinuous right part.