General inverse theorems for the best approximation in Banach spaces (Q2753327)

Classical theorems by Bernstein and Zygmund conclude the smoothness (i.e., differentiability and moduli of continuity) of a periodic function, \(f\), from the rapidity of the convergence to 0 of the distance from \(f\) to \(T_n\) the trigonometric functions of degree at most \(n\). To prove an analogous result in a Banach space, \(M\), this paper assumes the existence of a nested sequence of subspaces, \(M_n\), such that \(\cup M_n\) is dense in \(M\). The differentiable functions are replaced by functions in the domain of operators that map \(M_n\) into itself. An application is made to sequences of projections that are total, fundamental, and mutually orthogonal.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].