Nonlinear Approximation and Function Space of Dominating Mixed Smoothness
As the main object of this thesis we study a generalization of function spaces with dominating smoothness. The basic idea for this generalization is based on a splitting of the d variables into N groups of possibly varying length. In this way the usual spaces of dominating mixed smoothness as well as the classical isotropic spaces occur as special cases (for d groups of length one each or for 1 group of length d, respectively). More precisely, we will study first Sobolev-type spaces, and afterwards Besov- and Triebel-Lizorkin-type spaces are introduced. For these spaces several basic properties such as Fourier multipliers and duality are discussed. As the main tool for further studies characterizations by local means are proved. The main result of the first part of the thesis consists in a characterization by tensor product Daubechies wavelets. One immediate corollary of this characterization is the identification of certain Besov spaces of dominating mixed smoothness as tensor products of isotropic ones, establishing a connection to many recent discussions in high-dimensional approximation. The second part of this thesis is devoted to the study of one particular method of nonlinear approximation, m-term approximation with respect to the mentioned tensor product spaces in the framework of the spaces of dominating mixed smoothness. Here the wavelet characterization comes into play, allowing a reformulation of this problem using associated sequence spaces and the canonical bases. After some preparatory considerations, including the investigation of continuous and compact embeddings and duality, some explicit constructions for m-term approximation in several different settings are studied. Our main attention is turned on the asymptotic behaviour of certain worst case errors of this method. After reformulating the results from the explicit constructions in this sense, these results are extended using assertions about real interpolation and reiteration. Finally, the results on these aysmptotic rates are transferred back to function spaces, using once more the wavelet characterization. The results obtained in this way improve earlier ones by Dinh Dung and Temlyakov.