On the uniform modulus of continuity of certain discrete approximation operators (Q1114101)

For an interval \(I\subset\mathbb{R}\), continuous functions \(f\in C(I)\), the authors consider approximation operators \[ B^ f_ n(x) = \sum_{j\in J_ n} f(\xi_{jn}) p_{jn} (x) \] for \(x\in I\), \(n\in\mathbb{N}\), with \(J_ n\subseteq\mathbb{Z}\), \(\xi_{jn}\in I\) for \(n\in\mathbb{N}\), \(j\in J_ n\). The most important case of the main result reads: If the weights \(p_{jn}(x)\) satisfy the assumptions \[ p_{jn}(x)\geq 0,\quad \sum_{j\in J_ n} p_{jn}(x) \equiv 1,\quad 0<\mu_{2n}(x) = \sum_{j\in J_ n} (\xi_{jn} -x)^2 p_{jn} (x) < \infty, \] and \(p_{jn}(x)\in C_ 1(I)\) with \(p'_{jn}(x)\mu_{2n}(x)=p_{jn}(x)(\xi_{jn}-x)\), for \(x\in\overset\circ I\), then the authors prove that for the modulus of continuity \(\omega (B^ f_ n;h)\leq 4\omega(f,h),\) for all \(h\geq 0\), \(n\in\mathbb{N}\). Examples of operators where this theorem may be applied are especially those that are based on a probability distribution, e.g., the Bernstein polynomials, Szàsz-Mirakjan operators, Meyer-König-Zeller operators, and the discrete Favard operators.