arXiv subjects
Publications and source records attributed to Winfried Sickel.
We study the existence of the product of two weighted modulation spaces. For this purpose we discuss two different strategies. The more simple one allows transparent proofs in various situations. However, our second method allows a closer look onto associated norm inequalities under restrictions in the Fourier image. This will give us the opportunity to treat the boundedness of composition operators.
We compare Besov spaces with isotropic smoothness with Besov spaces of dominating mixed smoothness. Necessary and sufficient conditions for continuous embeddings will be given.
Motivated by classical results for Gevrey spaces and their applications to nonlinear partial differential equations we define so-called Gevrey-modulation spaces. We establish analytic as well as non-analytic superposition results on Gevrey-modulation spaces. These results are extended to a special weighted modulation space where the weight increases stronger than any polynomial but less than as in the Gevrey case.
In this article, the authors study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e.\,g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the $\pm$-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasize is given to the quasi-Banach case and to the interpolation property.
Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_\infty$-approximation, with precise control of all involved constants. As an illustration, we derive some results for periodic isotropic Sobolev spaces $H^s ({\mathbb T}^d)$ and Sobolev spaces of dominating mixed smoothness $H^s_{\rm mix} ({\mathbb T}^d)$, always equipped with natural norms. Some results for isotropic as well as dominating mixed Besov spaces are also obtained.
In this paper we investigate the asymptotic behaviour of Weyl numbers of embeddings of tensor product Besov spaces into Lebesgue spaces. These results will be compared with the known behaviour of entropy numbers.
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.
We study complex interpolation of weighted Besov and Lizorkin-Triebel spaces. The used weights $w_0,w_1$ are local Muckenhoupt weights in the sense of Rychkov. As a first step we calculate the Calderón products of associated sequence spaces. Finally, as a corollary of these investigations, we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin-Triebel spaces on $\R^d$.
In this paper we consider smoothness and decay properties of radial functions belonging to smoothness spaces related to Morrey spaces (Sobolev-Morrey spaces, Besov-type spaces and Besov-Morrey spaces). Within this framework we prove generalizations of the Radial Lemma of Strauss.
We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other words, we study the approximation numbers of the Sobolev embeddings $H^s_{\rm mix}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$, with particular emphasis on the dependence on the dimension $d$. For any fixed smoothness $s>0$, we find the exact asymptotic behavior of the constants as $d\to\infty$. We observe super-exponential decay of the constants in $d$, if $n$, the number of linear samples of $f$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. We present some surprising results for the so-called ``preasymptotic'' decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.
We continue our earlier investigations of radial subspaces of Besov and Lizorkin-Triebel spaces on $\R^d$. This time we study characterizations of these subspaces by differences.
We deal with decay and boundedness properties of radial functions belonging to Besov and Lizorkin-Triebel spaces. In detail we investigate the surprising interplay of regularity and decay. Our tools are atomic decompositions in combination with trace theorems.
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
We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings.
We study the optimal approximation of the solution of an operator equation Au=f by linear and nonlinear mappings. We identify those cases where optimal nonlinear approximation is better than optimal linear approximation.