In this paper we introduce the $a$-tempered integral and the $a$-tempered derivative for which almost all results from [5] can be simply transferred. Using the special sequences of $a$-tempered integrals and and $a$-tempered derivatives, from the point of view of sequential approach, we characterize some subspaces of $D'$. These spaces are of $K\{M_p\}$-type [2] for the special sequences $\{M_p\}$.