Winfried Sickel

Glenn Byrenheid, Dinh Dũng, Winfried Sickel et al.

We investigate the rate of convergence of linear sampling numbers of the embedding $H^{α,β} (\mathbb{T}^d) \hookrightarrow H^γ(\mathbb{T}^d)$. Here $α$ governs the mixed smoothness and $β$ the isotropic smoothness in the space $H^{α,β}(\mathbb{T}^d)$ of hybrid smoothness, whereas $H^γ(\mathbb{T}^d)$ denotes the isotropic Sobolev space. If $γ>β$ we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on "energy-norm based sparse grids" for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding $H^α_{mix} (\mathbb{T}^d) \hookrightarrow H^γ_{mix}(\mathbb{T}^d)$ and achieve optimality for Smolyak's algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding $H^α_{mix} (\mathbb{T}^d) \hookrightarrow L_q(\mathbb{T}^d)$ for $2<q\leq \infty$ where again Smolyak's algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation $β= γ$ (including the embedding into $L_2(\mathbb{T}^d)$). The best what we could prove there is a (probably) non-sharp results with a logarithmic gap between lower and upper bound.

Stephan Dahlke, Erich Novak, Winfried Sickel

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 domains