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We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.

In this paper we give the direct, inverse and equivalence theorem for Baskakov-Kantorovich-Bézier operators in the space $L_p[0,\infty)$ ($1 \leq p \leq \infty$) with Ditzian-Totik modulus.

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.

Let $A$ be the generator of an $n$-times integrated semigroup $T(\cdot)$ and let $r \in \mathbb{N}$. We first prove the equivalence of Riemann, Peano, and Taylor operators, which are three different expressions of the $r$-th power of $A_1$, the part of $A$ in the closure of the domain $D(A)$ of $A$. Then we discuss optimal and non-optimal rates of approximation of $T(\cdot)x$ for $x \in D(A^{r−1}_{1})$, via the $(n+r)$-th Taylor expansion of $T(\cdot)$ in terms of $A^k_1$, $k = 0, \ldots, r−1$.

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.