Recent developments in the theory of function spaces with dominating mixed smoothness. (English) Zbl 1289.46056

Rákosník, Jiří (ed.), Nonlinear analysis, function spaces and applications. Vol. 8. Proceedings of the spring school, NAFSA 8, Prague, Czech Republic, May 30–June 6, 2006. Prague: Czech Academy of Sciences, Mathematical Institute (ISBN 978-80-85823-50-9). 145-204 (2007).
This is a survey paper corresponding to a talk presented at the Spring School for Nonlinear Analysis, Function Spaces and Applications in 2006. It concerns the author’s joint work with W. Sickel and M. Krbec as well as the work which has been done by J. Vybíral within his thesis. The first goal is to discuss the Fourier-analytical approach to the theory of function spaces of Sobolev type, equivalent characterizations of these spaces with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis, approximation with respect to hyperbolic crosses, embeddings and traces is studied. This work follows results from M. Krbec and H.-J. Schmeisser [Ric. Mat. 45, 423–447 (1996; Zbl 0930.46029); Proc. R. Soc. Edinb., Sect. A, Math. 131, No. 3, 667–700 (2001; Zbl 0985.46018); Stud. Math. 181, No. 3, 255–284 (2007; Zbl 1131.46026)], M. C. Rodriguez Fernández [“Über die Spur von Funktionen mit dominierenden gemischten Glattheitseigenschaften auf der Diagonale.” Thesis, Jena (1997)], H.-J. Schmeisser and H. Triebel [Topics in Fourier analysis and function spaces. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. (1987; Zbl 0661.46024), or Chichester: John Wiley & Sons (1987; Zbl 0661.46025)], H.-J. Schmeisser and W. Sickel [J. Approximation Theory 128, No. 2, 115–150 (2004; Zbl 1045.41009)], J. Vybíral and W. Sickel [Czech. Math. J. 57, No. 4, 1239–1273 (2007; Zbl 1174.42027)] and J. Vybíral [“Characterization of function spaces with dominating mixed smoothness properties.” Jenaer Schriften Math. u. Inf., Math/Inf/15/03, 1–42 (2003); Funct. Approximatio Comment. Math. 33, 101–120 (2005; Zbl 1186.46033); Function spaces with dominating mixed smoothness. Diss. Math. 436, 1–73 (2006; Zbl 1101.46023)].
For the entire collection see [Zbl 1180.46003].
Reviewer: Petr Gurka (Praha)


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
65T60 Numerical methods for wavelets
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