Recent developments in the theory of function spaces with dominating mixed smoothness. (Q2846674)

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 \textit{M. Krbec} and \textit{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)], \textit{M. C. Rodriguez Fernández} [``Über die Spur von Funktionen mit dominierenden gemischten Glattheitseigenschaften auf der Diagonale.'' Thesis, Jena (1997)], \textit{H.-J. Schmeisser} and \textit{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)], \textit{H.-J. Schmeisser} and \textit{W. Sickel} [J. Approximation Theory 128, No. 2, 115--150 (2004; Zbl 1045.41009)], \textit{J. Vybíral} and \textit{W. Sickel} [Czech. Math. J. 57, No. 4, 1239--1273 (2007; Zbl 1174.42027)] and \textit{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)].NEWLINENEWLINEFor the entire collection see [Zbl 1180.46003].