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Schmeisser, Hans-Jürgen
Recent developments in the theory of function spaces with dominating mixed smoothness. (English). In: Rákosník, Jiří (ed.): Nonlinear Analysis, Function Spaces and Applications, Proceedings of the Spring School held in Prague, May 30-June 6, 2006. Vol. 8. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, 2007. pp. 145-204
MSC: 41A46, 42B25, 42B35, 42C40, 46E35, 47B06, 65T60
Keywords:
Function spaces; Fourier analysis; dominating mixed smoothness; hyperbolic cross approximation; local means; atoms; wavelets; embeddings; traces; entropy numbers
Summary:
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 with respect to hyperbolic crosses, embeddings and traces. We follow [42], [43], [44], [59], [63], [64], [70], and [94], [95], [96]. Partial results can be found also in [6], [7], [8], [37] and [48].
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