A generalized frame-work for inverse and equivalence results in approximation theory (Q2739390)

Extending the researches of \textit{Z. Ditzian} [J. Approximation Theory 62, No. 3, 316-339 (1990; Zbl 0728.41009)], \textit{M. Becker} and \textit{R. J. Nessel} [J. Approximation Theory 23), 99-103 (1978; Zbl 0388.41010)], and others, the author proves inverse and equivalence theorems concerning the approximation in Banach spaces. A prototype of the kind of results received by the author earlier [J. Approximation Theory 77, No.~2, 153-166 (1994; Zbl 0809.41025)] and related with the basic Jacksonian estimate \( A_\sigma (f)_{L_q}\leq c_k\omega_k (f;{1\over \sigma })_{L_q} \) of error in best approximation of \(f\in L_q(-\infty ,\infty),1 \leq q\leq \infty \), by integral functions of degree \(\sigma \leq 0 \),where \( \omega _k(f;t)_{L_q} (k\leq 1)\) is the \(k\)-th order integral modulus of smoothness of \(f\). Under the generalization the space \(L_q \) is replaced by a Banach space \(X \) in which the generalized Bernstein inequality for some subspaces \(X_n\subset X \) is true. \(\omega _k \) is replased by the Peetre's \(K- \)functional corresponding to the triple \(\{X,Y,\Phi \} \), where \(\Phi \) is a seminorm and \(Y=\{ f\in X: \Phi (f)<\infty \} \). The problem is investigated not only for a sequence of best approximations, but for a sequence of linear operators acting in \(X \). Applications of the obtained results to approximation theory in some Banach spaces and moduli of smoothness are given.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00032].