Tractability of tensor product problems in the average case setting (Q544118)

\textit{E. Novak} and \textit{H. Woźniakowski} addressed 91 open problems in their book [Tractability of multivariate problems. Volume I: Linear information. EMS Tracts in Mathematics 6. Zürich: European Mathematical Society (EMS) (2008; Zbl 1156.65001)], and the authors solve Open Problem 28: ``Weak tractability for linear tensor product problems in the average case setting''. That is, the authors derive a necessary and sufficient condition for a linear tensor product problem \(S = \{S_{d}\}\) in the average case setting to be weakly tractable, where \(S_{d}\)'s are linear operators and \(d\) is the number of variables in a multivariate problem. The authors show that \(S\) is weakly tractable iff \(\sum_{j>n}\lambda_{j} = o(\ln^{-2}n)\), where \(\lambda_{j}\)'s are eigenvalues. In particular, \(S\) is weakly tractable iff \(\ell = \lim_{j \rightarrow \infty}\lambda_{j} j \ln^{3} (j+1) = 0\).