Bernstein type inequalities associated with some differential operators
Let \(P(D): = \sum_{i=0}^r\alpha _i(x)(x(1-x))^iD^{r+i} \) denote the differential operator, where \(\alpha_i \in \)Lip \(\delta , 0<\delta \leq 1, \alpha_r(x)\neq 0\) for \(x\in [0,1], \phi (x): = \sqrt {x(1-x)}, \) and \(w_{\alpha , \beta} :=x^{\alpha} (1-x)^\beta . \) For an algebraic polynomial \(P_n \) of degree \(n\) and some \(\alpha , \beta \) the quantity \(\| w_{\alpha, \beta } \phi^{2r+1}P_n^{(2r+1)}\|_{L_p[0,1]}\) is estimated from above by \(\| w_{\alpha , \beta} P(D)P_n\|_{L_p[0,1]}\) and \(\| w_{\alpha , \beta} P_n\|_{L_p[0,1]}.\) In case \(r=0\) from the obtained estimate follows the well-known Bernstein type inequality [\textit{Z. Ditzian} and \textit{V. Totik}, ``Moduli of smoothness'' (1987; Zbl 0666.41001)]. Besides, the regularity of the operator \(P(D)\) with respect to all, but at most \(2r\) different \(L_p\) norms are established.
scientific article; zbMATH DE number 1117626