Best polynomial and Durrmeyer approximation in \(L_ p(S)\)
For a simplex \(S\subseteq R^ d\) and \(f\in L_ p(S)\) let \(E_ n(f)_{L_ p(S)}=\inf\{\| P_ n-f\|_ p:\) \(P_ n\) a polynomial of total degree at most \(n\)\} where \(\|\cdot\|_ p\) stands for the norm \(\|\cdot\|_{L_ p(S)}\) on \(L_ p(S)\), \(1\leq p\leq\infty\). As the Bernstein operator is not bounded on \(L_ p(S)\) it is the Kantorovich-Bernstein operator that can be compared with the best \(L_ p\)-approximation operator. But more suitable for this comparison is the Durrmeyer-Bernstein polynomial \(M_ nf\), which have also a series of nice properties --- it commutes with other operators of the same family and with an appropriate differential operator associated to it, it is self-adjoint and can be given by an expansion of orthogonal polynomials. For instance, \[ \|(M_ n-I)^ r f\|_ p\leq C n^{- r}\cdot\sum_{1\leq k\leq\sqrt n} k^{2r-1}\cdot E_ k(f)_ p. \] As a consequence of the results proved by the authors one obtains that for \(0<\alpha<1\), and \(1\leq p\leq\infty\), \(E_ n(f)_{L_ p(S)}=O(n^{- 2\alpha})\Leftrightarrow\| M_ n f-f\|_ p=O(n^{-\alpha})\). Also, for \(0<\alpha<r\), \(1\leq p\leq\infty\), \(\|(M_ n-I)^ r f\|_ p=O(n^{-\alpha})\Leftrightarrow E_ n(f)_ p=O(n^{- 2\alpha})\Leftrightarrow\omega^{2r}_ S(f,t)=O(t^{2\alpha})\). The relation with the differential operator \(P(D)\) associated to \(M_ n f\) is also considered: For an integer \(m\), \(\|(M_ n-I)^ m f\|_ p=O(n^{-m})\Leftrightarrow P(D)^ m f\) exists in the weak sense and belongs to \(L_ p(S)\), for \(1<p\leq\infty\) and to \({\mathcal M}(S)\) (the measures on \(S\)) for \(p=1\).
scientific article; zbMATH DE number 60866