On the adaptive and continuous information problems (Q1263941)

The paper deals with the adaption issue, showing that it can help for linear problems. For such problems (in a Banach space setting), the infimum of the ratio of adaptive and nonadaptive information is determined and related results are derived: let a and b respectively denote this infimum and the same infimum over all linear problems with Hilbert space range; then \(1/2\leq a\leq \sqrt{.8665}\) and \(\sqrt{2}/2<b<\sqrt{.8665}.\) Similar results are derived for finite- dimensional and \(L^ p\) range spaces. It is also shown that continuous information yields a smaller radius of information (error) than linear information in a Banach space.