Regularization of distributions

Sufficient conditions are presented for the construction of a regularization of a distribution in the form $a(\sigma)f$, where $f$ is a distribution and $a(\sigma)$ is an infinitely differentiable function outside a closed set $N$ which has power-like singularities of derivatives on $N$. Applications of such regularizations to an effective construction of solutions of the equation $Pu=f$, where $P(\sigma)$ is a polynomial, are considered. Bibliography: 14 titles.