Negative characters on the degree of the best approximation in Banach spaces (Q2730978)

This is a continuation of a previous article by the same author. Here there are obtained negative results on the degree of the best approximation related with a total fundamental sequence \(\{P_j\}_{j\in{\mathbb Z}}\) of mutually orthogonal projections in a Banach space \(X\). Let \(M_n=\) span\(\{P_j(X):|j|\leq n\}\subset X\), and \(E_n(f)\) is the best approximation of \(f\) from \(M_n\) in \(X\). A typical result: For every natural \(n\) let \(L_n\) be a bounded linear operator of \(X\) in \(M_n\). For any nonnegative continuous function on \([0,\infty)\) with \(\rho(0)=0\) the inequality \(\|L_n(f)-f\|\leq \rho(E_n(f))\) can not hold for all \(f\in X,\;n\in{\mathbb N}\). Applications for abstract Fourier expansions in Banach spaces and homogeneous Banach spaces, including certain classical function spaces are presented .