Pointwise estimate for the linear combinations of Bernstein-Durrmeyer operators (Q2749016)

Using Ditzian-Totik moduli of smoothness (pointwise moduli of smoothness) \(\omega^{2r}_{\varphi^\lambda} (f,t)\) the authors research the approximation by linear combinations of \(r\) order of Bernstein-Durrmeyer operators, the results of the authors involve many known results on classical moduli of smoothness and Ditzian-Totik moduli. The Bernstein-Durrmeyer operators are defined by \(L_n(f,x): =\sum^n_{k=0} P_{n,k}(x) (n+1)\int^1_0 f(t)P_{n,k}(t)dt\), NEWLINE\[NEWLINEP_{n,k}(x): ={n\choose k}x^k(1-x)^{n-k}, \quad 0\leq x\leq 1.NEWLINE\]NEWLINE Linear combinations of \(r\) order are defined by NEWLINE\[NEWLINEL_{n,r}(f,x): =\sum^{r-1}_{i=0} a_i (n)L_{n_i} (f,x).NEWLINE\]NEWLINE The author proves two theorems: Theorem 1.1: If \(f\in C[0, 1]\), \(r\in\mathbb{N}\), \(0<\alpha <{2r\over 2-\lambda}\), \(1-{1\over r}\leq \lambda\leq 1\), then NEWLINE\[NEWLINE\bigl|L_{n,r} (f,x)-f(x) \bigr|=O \biggl(\bigl(n^{-1/2} \delta_n^{1- \lambda}(x) \bigr)^\alpha \biggr) \Leftrightarrow \omega^{2 r}_{\varphi^\lambda} (f,t)=O (t^\alpha),NEWLINE\]NEWLINE here \(\varphi(x)= \sqrt{x(1-x)} (0\leq x\leq 1)\), \(\delta_n(x) =\varphi(x)+ {1\over\sqrt n}\).