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This paper is primarily concerned with approximation at 0 of an $(a,k)$-regularized resolvent family $R(\cdot)$ for Volterra integral equation. We shall consider convergence rates of some kind of local means $Q_m(t)$, $t \geq 0$, $m \geq 0$, of $R(t)/k(t)$. Some approximation theorems and local ergodic theorems with rates will be deduced from general approximation theorems for regularized approximation processes.

A-ergodic nets and A-regularized approximation processes of operators are introduced and their convergence theorems are discussed. There are strong convergence theorems, uniform convergence theorems, theorems on optimal convergence, and theorems on non-optimal convergence and its sharpness. The general results provide unified approaches to investigation of convergence rates of ergodic limits and approximation of various operator families. In particular, we shall deduce some results for an r-times integrated resolvent family for a Volterra integral equation. The latter contains integrated semigroups and integrated cosine functions as special cases.