On a certain class of approximation operators (Q2755005)

The sequence \(p=\{p_n\}_{n=0}^{\infty}\) of non zero polynomials is called of binomial type if it satisfies the following identity: NEWLINE\[NEWLINEp_n(x+y)=\sum_{k=0}^n {n!\over k!(n-k)!}p_k(x)p_{n-k}(y); \hbox{for all} \;x,y\in\mathbb{R}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEGiven an analytic function \(H\), whose power series is invertible, the author defines a sequence \(p = \{p_n\}_{n\geq 0}\) of polynomials which depends on \(H\) and it is, under certain aditional hyphotheses, of binomial type. The author defines and studies the approximation properties of the Kantorovich type operators associated to the sequence \(p\), defined by: NEWLINE\[NEWLINE(K^H_n f)(x)= (n+1)\sum_{k=0}^n p_{n,k}\int_{{k\over n+1}}^{{k+1\over n+1}}f(t)dt; \;\hbox{for} \;f\in L_1(0,1);NEWLINE\]NEWLINE where \(p_{n,k}(x)={1\over p_n(1)}{n!\over k!(n-k)!}p_k(x)p_{n-k}(1-x)\). In the paper several direct theorems are given on the errors of best approximation with these operators. The most important tools for the proofs are: the representation of these operators as integral operators, and the use of certain known equivalences of moduli of smoothness and K-functionals. The paper is well-written and (from the point of view of the reviewer) of interest, but contains certain unproved claims.