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Winfried Sickel

Publications and source records attributed to Winfried Sickel.

At least 19 recordsLinked to original sources
arXiv2026

QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces

We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter $α>0$. We study the worst case error of integration over the norm unit ball and provide upper error bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary $(t,m,s)$-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary $(t,m,s)$-nets as sample points yields the best possible rate of convergence. Afterwards we study spaces of integrands of fractional smoothness $α\in (0,1)$ and state a sharp Koksma-Hlawka-type inequality. More precisely, we show that on those spaces the worst case error of integration is equal to the corresponding fractional discrepancy. Those spaces can be continuously embedded into tensor product Bessel potential spaces, also known as Sobolev spaces of dominated mixed smoothness, with the same set of parameters. The latter spaces can be embedded into suitable Besov spaces of dominating mixed smoothness $α$, which in turn can be embedded into the Haar wavelet spaces with the same set of parameters. Therefore our upper error bounds on Haar wavelet spaces for QMC cubatures based on $(t,m,s)$-nets transfer (with possibly different constants) to the corresponding spaces of integrands of fractional smoothness and to Sobolev and Besov spaces of dominating mixed smoothness. Moreover, known lower error bounds for periodic Sobolev and Besov spaces of dominating mixed smoothness show that QMC integration based on arbitrary $(t,m,s)$-nets yields the best possible convergence rate on periodic as well as on non-periodic Sobolev and Besov spaces of dominating smoothness.

math.NA↗
arXiv2024

Regularity of Characteristic Functions in Besov-Type and Triebel--Lizorkin-Type Spaces

In this article, the authors determine the optimal regularity of characteristic functions in Besov-type and Triebel--Lizorkin-type spaces under restrictions on the measure of the $δ$-neighborhoods of the boundary. In particular, the necessary and sufficient conditions for the membership in these spaces of characteristic functions of the snowflake domain and also some spiral type domains are obtained.

math.FA↗
arXiv2022

Pointwise Multipliers for Besov Spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with Only Logarithmic Smoothness

In this article, we establish a characterization of the set $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ of all pointwise multipliers of Besov spaces $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ with only logarithmic smoothness $b\in\mathbb{R}$ in the special cases $p=1$ and $p=\infty$. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions $e^{ik\cdot x}$ with $k\in\mathbb{Z}^n$ and $x\in\mathbb{R}^n$, are pointwise multipliers of $B^{0,b}_{1,\infty}(\mathbb{R}^n)$ and $B^{0,b}_{\infty,\infty}(\mathbb{R}^n)$, respectively; furthermore, we obtain the explicit estimates of $\|e^{ik \cdot x}\|_{M(B^{0,b}_{1,\infty}(\mathbb{R}^n))}$ and $\|e^{ik \cdot x}\|_{M(B^{0,b}_{\infty,\infty}(\mathbb{R}^n))}$. In the case that $p\in(1,\infty)$, we give some sufficient conditions and some necessary conditions of the pointwise multipliers of $B^{0,b}_{p,\infty}(\mathbb{R}^n)$ and a complete characterization of $M(B^{0,b}_{p,\infty}(\mathbb{R}^n))$ is still open. However, via a different method, we are still able to accurately calculate $\|e^{ik \cdot x}\|_{M(B^{0,b}_{p,\infty}(\mathbb{R}^n))}$, $k\in\mathbb{Z}^n$, in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration.

math.FA↗
arXiv2021

$s$-Numbers of Embeddings of Weighted Wiener Algebras

In this paper we study the asymptotic behavior of Kolmogorov, approximation, Bernstein and Weyl numbers of embeddings $ \mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to L_2(\mathbb{T}^d)$ and $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d) \to \mathcal{A}(\mathbb{T}^d)$, where $\mathcal{A}^{s,r}_{\rm mix}(\mathbb{T}^d)$ is a weighted Wiener algebra of mixed smoothness $s$ and $\mathcal{A}(\mathbb{T}^d)$ is the Wiener algebra itself, both defined on the $d$-dimensional torus $\mathbb{T}^d$. Our main interest consists in the calculation of the associated asymptotic constants.

math.FA↗
arXiv2020

How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings

We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert-Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension $d$. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into $L_2$ and $H^1$. In other words, we investigate the worst-case approximation error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for $n$. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Wo{ź}niakowski.

math.NA↗
arXiv2020

Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains

In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.

math.CA↗
arXiv2019

Properties of Kondratiev spaces

In this paper, we investigate Kondratiev spaces on domains of polyhedral type. In particular, we will be concerned with necessary and sufficient conditions for continuous and compact embeddings, and in addition we shall deal with pointwise multiplication in these spaces.

math.FA↗
arXiv2019

The Haar System in Besov-type Spaces

Some Besov-type spaces $B^{s,τ}_{p,q}(\mathbb{R}^n)$ can be characterized in terms of the behavior of the Fourier--Haar coefficients. In this article, the authors discuss some necessary restrictions for the parameters $s$, $τ$, $p$, $q$ and $n$ of this characterization. Therefore, the authors measure the regularity of the characteristic function $\mathcal X$ of the unit cube in $\mathbb{R}^n$ via the Besov-type spaces $B^{s,τ}_{p,q}(\mathbb{R}^n)$. Furthermore, the authors study necessary and sufficient conditions such that the operation $\langle f, \mathcal{X} \rangle$ generates a continuous linear functional on $B^{s,τ}_{p,q}(\mathbb{R}^n)$.

math.CA↗
arXiv2017

Pointwise Multipliers for Besov Spaces of Dominating Mixed Smoothness - II

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(\mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $p\leq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(\mathbb{R}^d)$.

math.FA↗
arXiv2017

Traces of anisotropic Besov--Lizorkin--Triebel spaces---a complete treatment of the borderline cases

Including the previously untreated borderline cases, the trace spaces in the distributional sense of the Besov--Lizorkin--Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the trace are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the previously untreated cases. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J.~Franke and B.~Jawerth to the anisotropic scales.

math.AP↗
arXiv2017

A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.

math.AP↗
arXiv2017

On the trace problem for Triebel--Lizorkin spaces with mixed norms

The subject is traces of Sobolev spaces with mixed Lebesgue norms on Euclidean space. Specifically, restrictions to the hyperplanes given by the first and last coordinates are applied to functions belonging to quasi-homogeneous, mixed-norm Lizorkin--Triebel spaces; Sobolev spaces are obtained from these as special cases. Spaces admitting traces in the distribution sense are characterised except for the borderline cases; these are also covered in case of the first variable. With respect to the first variable the trace spaces are proved to be mixed-norm Lizorkin--Triebel spaces with a specific sum exponent. For the last variable they are similarly defined Besov spaces. The treatment includes continuous right-inverses and higher order traces. The results rely on a sequence version of Nikolskij's inequality, Marschall's inequality for pseudo-differential operators (and Fourier multiplier assertions), as well as dyadic ball criteria.

math.AP↗
arXiv2016

Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms

This is a contribution to the theory of Lizorkin--Triebel spaces having mixed Lebesgue norms and quasi-homogeneous smoothness. We discuss their characterisation in terms of general quasi-norms based on convolutions. In particular, this covers the case of local means, in Triebel's terminology. The main step is an extension of some crucial inequalities due to Rychkov to the case with mixed norms.

math.FA↗
arXiv2016

Anisotropic Lizorkin--Triebel Spaces with Mixed Norms --- Traces on Smooth Boundaries

This article deals with trace operators on anisotropic Lizorkin--Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self-contained exposition of Lizorkin--Triebel spaces on manifolds and extend these results to mixed-norm Lizorkin--Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right-inverse of the trace is given explicitly and proved to be continuous in the scale of mixed-norm Lizorkin--Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.

math.AP↗