*(English)*Zbl 1064.41015

The author takes an array $\left\{{x}_{n,k}\right\}$, $n\ge 1$, $k\ge 0$ such that for each $k\ge 0$, ${\gamma}_{k}$ exists for which ${x}_{n,k}=O\left({n}^{-{\gamma}_{k}}\right)$, $x\to \infty $ (the reviewer could not find where the latter assumption is used or even mentioned again in the paper). And he takes a family of nonnegative functions ${\varphi}_{n,k}\in {C}^{1}\left({\mathbb{R}}_{+}\right)$, $n\ge 1$, $k\ge 0$, satisfying

and such that there exists a sequence of positive functions ${\psi}_{n}\in C\left({\mathbb{R}}_{+}\right)$, for which

He defines the operators

for all functions $f$ defined on ${\mathbb{R}}_{+}$ for which the sums converge. Special cases are the Szász-Mirakyan and the Baskakov operators. (Note that in the paper there is a mistake in the definition of the Baskakov operators.) The author investigates the properties of the $r$th central moment of ${L}_{n}$, namely,

not noticing that nothing in his assumptions guarantees the convergence of the infinite sum for $r\ge 2$. Thus without additional assumptions Lemma 1 and hence Theorem 1, are invalid. Assuming that we add the appropriate assumptions above, the reviewer fails to see the proof of Lemma 2 since the author indicates nothing about the behavior of ${{\Lambda}}_{n,2m-1}^{\text{'}}\left(x\right)$ that is needed in his induction step. In addition, the reviewer wonders why Lemma 2 “might be of interest in its own right”. The methods of proof are completely standard, notwithstanding the above comments.