Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces Hs(Td)Hs(Td) of fractional smoothness s>0s>0 for various equivalent norms including the classical one. The error is always measured in L2(Td)L2(Td). Particular emphasis is given to the dependence of all constants on the dimension dd. We capture the exact decay rate in nn and the exact decay order of the constants with respect to dd, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.