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We investigate the approximation of $d$-variate periodic functions in Sobolevspaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensionaltorus, where the approximation error is measured in the $L_2-$norm. In otherwords, we study the approximation numbers of the Sobolev embeddings $H^s_{\rmmix}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$, with particular emphasison the dependence on the dimension $d$. For any fixed smoothness $s>0$, we findthe exact asymptotic behavior of the constants as $d\to\infty$. We observesuper-exponential decay of the constants in $d$, if $n$, the number of linearsamples of $f$, is large. In addition, motivated by numerical implementationissues, we also focus on the error decay that can be achieved by low rankapproximations. We present some surprising results for the so-called``preasymptotic'' decay and point out connections to the recently introducednotion of quasi-polynomial tractability of approximation problems.

Approximation of mixed order Sobolev functions on the $d$-torus -- Asymptotics, preasymptotics and $d$-dependence