A Voronovskaja-type formula and preservation properties of a class of operators (Q2751673)

The first result on saturation was obtained by Voronovskaja in 1932 for the classical Bernstein operators \(B_n(f)= \sum^n_{i=0} f(i/n)C(n,i) x^i(1-x)^{n-i}\), where \(f\in C[0,1]\). An asymptotic formula of Voronovskaja-type (theorem 1) is established in this paper for integral operators \(P_n\) introduced by \textit{D. Mache} and \textit{D.-Z. Zhou} [J. Approximation Theory 84, No. 2, 145-161 (1996; Zbl 0840.41017)]. Let \(P_n\) be defined as \(P_nf(x)= \sum^n_{i=0} A_{in}(f) C(n,i)x^i (1-x)^{n-i}\), \(f\in C[0,1]\), where NEWLINE\[NEWLINEA_{in}(f)= B(ci+a+1,cn-ci+b+1)^{-1} \int^1_0 t^{ci+a} (1-t)^{cn-ci+b} f(t)dt,NEWLINE\]NEWLINE \(B\) is Euler's Beta function, \(c:=[n^\alpha]\), \(\alpha\geq 0\) and \(a,b\geq -1\) are fixed. Theorem 1: If \(\alpha=0\) then \(\lim_n n(P_nf(x) -f(x))=(a+1- (a+b+2)x)f' (x)+x(1-x)f''(x)\). If \(\alpha>0\) then \(\lim_n n(P_nf(x)- f(x))=x(1-x)f''(x)/2\). A preservation property of \(P_n\) which generalizes earlier results on the subject, is presented in the form of the following result. Theorem 2: Let \(0\leq m\leq n\) and let \(\{f_0, \dots f_m\}\subset C[0,1]\) be a Chebyshev system. Then \(\{P_nf_0, \dots,P_n f_m \}\) is also a Chebyshev system. If \(f\in C[0,1]\) is convex with respect to \(\{f_0, \dots,f_m\}\), then \(P_nf\) is convex with respect to \(\{P_nf_0, \dots,P_n f_m\}\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00041].