On the evaluation of remainders in some linear approximation formulas (Q2751677)

Using the usual notation for the Bernstein bases \(p_{n,j}(t)= {n\choose j}t^j(1-t)^{n-j}\), one can define this operator by the formula NEWLINE\[NEWLINE S^{\alpha, \beta}_{m,r,s} (f)(x)=\sum^{m-sr}_{k=0} p_{m-sr,k}(x)\left [\sum^s_{j =0} p_{s,j}(x)f \left({k+ \alpha+ jr\over m+\beta} \right)\right]NEWLINE\]NEWLINE where \(m\) is a natural number, \(\alpha\) and \(\beta \) are real parameters, such that \(0\leq \alpha \leq\beta\), while \(r\) and \(s\) are non-negative parameters, subject to the condition \(sr\leq m\). In this paper the authors investigate the remainder term of approximation formula NEWLINE\[NEWLINEf(x)=(S_{m,rs}f) (x)+(R^{\alpha, \beta}_{m,r,s}f)(x).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0968.00041].