Weighted simultaneous approximation by algebraic projection operators (Q1187297)

Let \(X=C[-1,1]\) endowed with the supremum norm \(\|\;\|\). An operator \(L_ n\) on \(X\) is called a polynomial projection operator on \(\rho_ n\), the set of algebraic polynomials of degree at most \(n\geq 0\) if \[ L_ nf\in\rho_ n(f\in X),\;L_ np_ n=p_ n(p_ n\in\rho_ n). \] In terms of the error of best approximation \[ E_ n(f)=\inf\{\| f-p_ n\|:p_ n\in\rho_ n\} \] it is well known that \(\| L_ nf-f\|\leq K\| L_ n\| E_ n(f)\) \((f\in X)\). The author gives an analogous result for the remainder of the simultaneous approximation of \(f\) by \(L_ nf\); i.e. for the \(r\)th derivative \((f-L_ nf)^{(r)}(x)\). The sharpness of the relevant estimates is discussed. The work is done on the weighted Sobolev space \(C^ r_ \varphi\) of functions \(f\) in \(X\) are \(r\) times differentiable on \((-1,1)\) such that \(\varphi^ rf^{(r)}\in X\) where \(\varphi(x)=\sqrt{1-x^ 2}(-1\leq x\leq 1)\) and \(C^ 0_ \varphi=X\). For earlier work of this type see \textit{R. O. Runck}, \textit{J. Szabados} and \textit{P. Vértesi} [Acta. Sci. Math. 53, No. 3/4 287-293 (1989; Zbl 0709.42001)].