A relation between Peano kernels and a theorem of Bernstein (Q1066387)

The authors are concerned with the following generalization of a theorem of S. N. Bernstein: let \(E_ n\) be a semi-norm which satisfies ''Property B'' (i.e.: \(| f^{(n+1)}(x)| \leq g^{(n+1)}(x)\) implies \(E_ n(f)\leq E_ n(g))\). Then, for each \(f\in C^{(n+1)}[a,b]\), there exists \(\xi\in (a,b)\) such that \(E_ n(f)=\frac{1}{(n+1)!}| f^{(n+1)}(\xi)| E_ n(x^{n+1})\) [cf. the authors, Approximation Theory III, Proc. Conf. Hon. G. G. Lorentz, Austin/Tex. 1980; 513-516 (1980; Zbl 0506.41010)]. Some relations between this theorem and Peano kernels are discussed in more detail.