The aim of these lectures is to present a survey of some results on spaces of functions with dominating mixed smoothness. These results concern joint work with Winfried Sickel and Miroslav Krbec as well as the work which has been done by Jan Vybíral within his thesis. The first goal is to discuss the Fourier-analytical approach, equivalent characterizations with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis we study approximation...

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{\pi ,q}^s$ on a regular domain of ${ℝ}^d.$ The result is: if s - d(1/π - 1/p)+> 0, then the Kolmogorov metric entropy satisfies H(ε) ~ ε-d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound,...

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space $B_{\pi ,q}^s$ on a regular domain of ${ℝ}^d.$ The result is: if $s-d{(1/\pi -1/p)}_{+}$$\>0,$ then the Kolmogorov metric entropy satisfies $H\left(ϵ\right)\sim {ϵ}^{-d/s}$. This...