## Spaces of functions of mixed smoothness and approximation from hyperbolic crosses.(English)Zbl 1045.41009

The authors study the relation between classes of functions defined by rates of best approximation with respect to hyperbolic crosses and smoothness spaces either defined by Fourier analytical tools or defined by differences with a dominating mixed term. Section 2 contains the definition, properties and some equivalent characterizations of Besov-Lizorkin-Triebel classes of dominating mixed smoothness. Section 3 deals with the description of approximation spaces with respect to the hyperbolic cross.A characterization as interpolation spaces is also studied. Section 4 collects some rsults on the interpolation of spaces of dominating mixed smoothness. The main contributions of the paper are contained in Section 5.Here the authors clarify the relations between the approximation classes on the one side and the spaces of Besov and Lizorkin-Triebel type on the other. Finally, in Section 6 some results regarding the approximation by partial sums with respect to hyperbolic crosses are presented.

### MSC:

 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A25 Rate of convergence, degree of approximation 41A63 Multidimensional problems 42B99 Harmonic analysis in several variables 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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