We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness s > 0 on the d-dimensional torus, where the approximation error is measured in the L 2 −norm. In other words, we study the approximation numbers of the Sobolev embeddings H, with particular emphasis on the dependence on the dimension d. For any fixed smoothness s > 0, we find the exact asymptotic behavior of the constants as d → ∞. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f , is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the socalled "preasymptotic" decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

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Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for L 2 -approximation of Sobolev functions into estimates for L ∞ -approximation, with precise control of all involved constants. As