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On the convergence of a truncated class of operators. (English) Zbl 1064.41015

The author takes an array {x n,k }, n1, k0 such that for each k0, γ k exists for which x n,k =O(n -γ k ), x (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 ϕ n,k C 1 ( + ), n1, k0, satisfying

k=0 ϕ n,k (x)=1, k=0 x n,k ϕ n,k (x)=x,x0,

and such that there exists a sequence of positive functions ψ n C( + ), for which

ψ n (x)ϕ n,k ' (x)=(x n,k -x)ϕ n,k (x)·

He defines the operators

L n (f,x):= k=0 ϕ n,k (x)f(x n,k ),n1,x0,

for all functions f defined on + 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 rth central moment of L n , namely,

Λ n,r (x):= k=0 ϕ n,k (x)(x-x n,k ) r ,

not noticing that nothing in his assumptions guarantees the convergence of the infinite sum for r2. 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 Λ n,2m-1 ' (x) 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:
41A36Approximation by positive operators
41A25Rate of convergence, degree of approximation