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We investigate the approximation of -variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness  on the -dimensional torus, where the approximation error is measured in the -norm. In other words, we study the approximation numbers  of the Sobolev embeddings , with particular emphasis on the dependence on the dimension . For any fixed smoothness , we find two-sided estimates for the approximation numbers as a function in  and . We observe super-exponential decay of the constants in , if , the number of linear samples of , is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small ). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence