A new characterization of (s,t)-weak tractability

Siedlecki and Weimar (2015) defined the notion of (s,t)-weak tractability for linear multivariate problems, which holds if the information complexity of the multivariate problem is not exponential in dt and ε−s, where d is the number of variables and ε is the error threshold with positive s and t. For Hilbert spaces, they were able to characterize (s,t)-weak tractability in terms of how quickly the corresponding ordered singular values decay. Using this result, they studied the embedding of Hr(Td) into L2(Td), where Td is the d-dimensional torus, determining precisely when this problem is (s,t)-tractable for a given  d and  r. Their proof is based on deep results of Kühn et al. (2014), which are complicated by the difficulty of ordering the singular values. In this paper, we provide a new characterization of (s,t)-weak tractability of multivariate problems over Hilbert spaces, which does not require us to order the singular values. This allows us to obtain a new, and somewhat simpler, proof of the Siedlecki and Weimar (2015) result that does not need to use the results of Kühn et al. (2014).