Function spaces of dominating mixed smoothness, Weyl and Bernstein numbers
Function spaces of dominating mixed smoothness were first introduced in the early sixties. Recently, there is an increasing interest in those spaces in information-based complexity and high-dimensional approximation. In this work, on the one hand, we concentrate on studying some further properties of Besov-Triebel-Lizorkin spaces of dominating mixed smoothness such as pointwise multiplication, characterization by mixed differences, and change of variable operators which are connected to numerous applications. On the other hand, we investigate the order of convergence of Weyl and Bernstein numbers of compact embeddings of tensor product Sobolev and Besov spaces into Lebesgue spaces on the unit cube. These quantities belong to the class so-called s-numbers and play an important role in the study of the complexity problems since they are lower bounds for worst-case approximation errors. Our method is based on the wavelet decomposition of Besov-Triebel-Lizorkin spaces of dominating mixed smoothness to reduce the problem to analyzing Weyl and Bernstein numbers in the level of sequence spaces.
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