## Spaces of functions of mixed smoothness from the decomposition point of view.(English. Russian original)Zbl 0707.46025

Proc. Steklov Inst. Math. 187, 163-184 (1990); translation from Tr. Mat. Inst. Steklova 187, 143-161 (1989).
This survey deals with the spaces $$S^ r_ pL$$ and $$S^{\alpha}_{p,\theta}B$$ of Sobolev and Besov type, respectively, having dominating mixed smoothness properties. The authors describe decomposition procedures, characterizations via approximation and Fourier-analytical methods of Paley-Littlewood type. Finally the peiodic case is considered.
Reviewer: H.Triebel

### MSC:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)