New Preasymptotic Estimates for Approximation of Periodic Sobolev Functions

Approximation of Sobolev embeddings is a well-studied subject in high-dimensional approximation, with many application to different branches of mathematics. E.g., for isotropic Sobolev spaces $$ H^{s} \left( {{\mathbb{T}}^{d} } \right) $$ of fractional smoothness s > 0 on the d-dimensional torus it is known that the approximation numbers an of the embedding $$ H^{s} \left( {{\mathbb{T}}^{d} } \right) \hookrightarrow L_{2} \left( {{\mathbb{T}}^{d} } \right) $$ behave like $$ a_{n} \sim n^{{{{ - s} \mathord{\left/ {\vphantom {{ - s} d}} \right. \kern-0pt} d}}} $$ as $$ n \to \infty $$, where the (weak) equivalence $$ \sim $$ holds only up to multiplicative constants which are not known explicitly. However, for practical purposes it is more relevant to know the preasymptotic behaviour of the an, i.e. for small n, say $$ n \le 2^{d} $$. In this range the dependence on n is only logarithmic. The main results in this note are sharp two-sided preasymptotic estimates for approximation of isotropic Sobolev functions on $$ {\mathbb{T}}^{d} $$. In particular we give explicit constants, which show the exact dependence on the dimension d, the smoothness s, and further parameters of the norm. This improves the known results in the literature. Moreover, we prove a new preasymptotic estimate for approximation of Sobolev functions of dominating mixed smoothness.