Winfried Sickel

Morrey and Campanato meet Besov, Lizorkin and Triebel

During the last 60 years the theory of function spaces has been a subject of growing interest and increasing diversity. Based on three formally different developments, namely, the theory of Besov and Triebel-Lizorkin spaces, the theory of Morrey and Campanato spaces and the theory of Q spaces, the authors develop a unified framework for all of these spaces. As a byproduct, the authors provide a completion of the theory of Triebel-Lizorkin spaces when p =

Optimal approximation of elliptic problems by linear and nonlinear mappings I

We study the optimal approximation of the solution of an operator equation Au=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs on an arbitrary bounded Lipschitz domain. Here we prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. Our results are concerned with approximation, not with computation. Our goal is to understand better the possibilities of nonlinear approximation.

Entropy Numbers of Embeddings of Weighted Besov Spaces

AbstractWe investigate the asymptotic behavior of the entropy numbers of the compact embedding

Hölder Inequalities and Sharp Embeddings in Function Spaces of

B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}).

and

Here

F^s_{pq}

B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)

Type

denotes a weighted Besov space, where the weight is given by

Best m-term approximation and Lizorkin–Triebel spaces

w_\alpha (x) = (1+| x |^2)^{\alpha/2}