Linear average and stochastic n-widths of Besov embeddings on Lipschitz domains

AbstractIn this paper, we determine the asymptotic degree of the linear average and stochastic n-widths of the compact embeddings Bq0s+t(Lp0(Ω))↪Bq1s(Lp1(Ω)),t>max{d(1/p0−1/p1),0},1≤p0,p1,q0,q1≤∞, where Bq0s+t(Lp0(Ω)) is a Besov space defined on the bounded Lipschitz domain Ω⊂Rd

Bernstein Numbers of Embeddings of Isotropic and Dominating Mixed Besov Spaces

The purpose of the present paper is to investigate the decay of Bernstein numbers of the embedding from $B^t_{p_1,q}((0,1)^d)$ into the space $L_{p_2}((0,1)^d)$. The asymptotic behaviour of Bernstein numbers of the identity $id: S_{p_1,p_1}^tB((0,1)^d)\rightarrow L_{p_2}((0,1)^d)$ will be also considered.Comment: 31 pages, 1 figur

Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings III: Frames

We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable nonlinear frame widths. The main advantage of frames compared to Riesz basis, which were studied in our earlier papers, is the fact that we can now handle arbitrary bounded Lipschitz domains--also for the upper bounds. Key words: elliptic operator equation, worst case error, frames, nonlinear approximation, best n-term approximation, manifold width, Besov spaces on Lipschitz domainsComment: J. Complexity, to appear. Final version, minor mistakes correcte

Almost diagonal matrices and Besov-type spaces based on wavelet expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces $b^\alpha_{p,q}(\nabla)$ of Besov type is developed. Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive methods, non-linear approximation, manifolds, domain decomposition.Comment: 38 pages, 2 figure

Besov regularity for operator equations on patchwise smooth manifolds

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $\Omega$.Comment: 42 pages, 3 figures, updated after peer review. Preprint: Bericht Mathematik Nr. 2013-03 des Fachbereichs Mathematik und Informatik, Universit\"at Marburg. To appear in J. Found. Comput. Mat

Besov regularity of solutions to the p-Poisson equation

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau = \sigma/d+1/p$, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to $p$-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure

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