Ultrastability of \(n\)th minimal errors (Q432764)

The author applies ultraproducts (the techniques from the local theory of Banach spaces) to information-based complexity theory in order to understand the local properties of \(n\)th minimal errors. He studies this issue in the deterministic setting with adaptive information consisting of linear functionals. A stability property of the \(n\)th minimal errors with respect to ultraproducts is investigated. The author presents the analysis for nonlinear continuous operators defined on open sets after introducing a suitable generalization of the ultraproduct of linear operators to this nonlinear situation.NEWLINENEWLINEThe question of \textit{A. Hinrichs, E. Nowak} and \textit{H. Woźniakowski} [``Discontinuous information in the worst case and randomized settings'', Math. Nachr., \url{doi:10.1002/mana.201100128}] (``whether the \(n\)th minimal error of a continuous operator is the supremum of the \(n\)th minimal errors of all its restrictions to finite dimensional subspaces?'') is answered negatively showing that the \(n\)th minimal errors considered are neither regular nor maximal. However, the author also shows that a positive answer can be expected if the operator is compact or the target space is 1-complemented in its bidual. The author shows that the \(n\)th minimal errors are \(s\)-numbers the linear case as well.