Weak type inequalities for best simultaneous approximation in Banach spaces (Q1963958)

In this paper, the author further generalizes results by \textit{P. L. Butzer} and \textit{K. Scherer} on the characterization of the approximation order of best approximation in arbitrary Banach spaces [Aequationes Math. 3, 170-185 (1969; Zbl 0179.09101)]. His results now allow the inclusion of characterizations of weighted best algebraic approximation in terms of the main part modulus of \textit{Z. Ditzian} and \textit{V. Totik} [Moduli of smoothness (1987; Zbl 0666.41001)] as a special case. The key ingredients for the characterizations are weak-type versions of the Jackson and Bernstein inequalities and an equivalence condition for certain norms and seminorms on subspaces.