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On the convergence of a truncated class of operators. (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 ${\varphi }_{n,k}\in {C}^{1}\left({ℝ}_{+}\right)$, $n\ge 1$, $k\ge 0$, satisfying

$\sum _{k=0}^{\infty }{\varphi }_{n,k}\left(x\right)=1,\phantom{\rule{1.em}{0ex}}\sum _{k=0}^{\infty }{x}_{n,k}{\varphi }_{n,k}\left(x\right)=x,\phantom{\rule{1.em}{0ex}}x\ge 0,$

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

${\psi }_{n}\left(x\right){\varphi }_{n,k}^{\text{'}}\left(x\right)=\left({x}_{n,k}-x\right){\varphi }_{n,k}\left(x\right)·$

He defines the operators

${L}_{n}\left(f,x\right):=\sum _{k=0}^{\infty }{\varphi }_{n,k}\left(x\right)f\left({x}_{n,k}\right),\phantom{\rule{1.em}{0ex}}n\ge 1,\phantom{\rule{1.em}{0ex}}x\ge 0,$

for all functions $f$ defined on ${ℝ}_{+}$ 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,

${{\Lambda }}_{n,r}\left(x\right):=\sum _{k=0}^{\infty }{\varphi }_{n,k}\left(x\right){\left(x-{x}_{n,k}\right)}^{r},$

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.

##### MSC:
 41A36 Approximation by positive operators 41A25 Rate of convergence, degree of approximation