Bernstein polynomials and modulus of continuity
The author describes several properties related to the first order modulus of continuity, which are preserved by the operator given by Bernstein polynomials. Let the function \(\omega(t)\) on \([0,1]\) be a modulus of continuity. By \(H^\omega\) we denote the class of continuous functions on \([0,1]\) satisfying the inequality \(\omega(f,t)\leq\omega(t)\). The author proves: (iii)* if \(\omega(t)\) is a modulus of continuity, then \(f\in H^\omega\) implies that for all \(n\geq 1\), \(B_n(f;x)\in H^{2\omega}\); if \(\omega(t)\) is concave (upper convex), then \(f\in H^\omega\) implies that for all \(n\geq 1\), \(B_n(f;x)\in H^\omega\). (iv) if \(\omega(t)\) is a modulus of continuity, then for each \(n\geq 1\), \(B_n(\omega;t)\) is also a modulus of continuity and \(B_n(\omega;t)\leq 2\omega(t)\); if \(\omega(t)\) is concave, then for each \(n\geq 1\), \(B_n(\omega;t)\) is a concave modulus of continuity and \(B_n(\omega;t) \leq\omega (t)\). (v) if \(f (x)\) is a non-negative function such that \(x^{-1}f(x)\) is non-increasing on \([0, 1]\), then for each \(n\geq 1\), \(x^{-1} B_n(f;x)\) is non-increasing also. The author gives some examples. Remark of reviewer. For generalization of (iii)* see \textit{G. A. Anastassiou}, and \textit{C. Cottin}, and \textit{H. H. Gonska} [Analysis 11, No. 1, 43-57 (1991; Zbl 0722.41021)] proved: \(\omega(B_nf; \delta)\leq 1\cdot \omega^* (f;\delta)\), \(\delta\geq 0\), here the constant 1 is best possible, \(\omega^* (f;\delta)\) is the least concave majorant of \(\omega(f; \delta)\) (we have \(\omega(f; \delta)\leq \omega^* (f;\delta)\leq 2\omega (f; \delta))\). If \(\omega(t)\) is a moduls of continuity and \(f\in H^\omega\), we have \(\omega(B_nf; \delta)\leq\omega^* (f;\delta)\leq 2\omega (f;\delta)\leq 2\omega (\delta)\), hence for all \(n\geq 1\) \(B_nf\in H^{2\omega}\). If \(\omega(t)\) is concave, then \(f\in H^\omega (\omega (f;t)\leq \omega(t))\) implies that \(\omega^* (f;t) \leq\omega (t)\) and \(\omega(B_nf; t)\leq 1\cdot\omega^* (f;t)\leq \omega(t)\), hence for all \(n\geq 1\), \(B_nf\in H^\omega\).
scientific article; zbMATH DE number 1417557