Related papers: Embeddings for Infinite-Dimensional Integration an…
QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic…
In this paper embeddings between weighted Copson function spaces ${\operatorname{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Ces\`{a}ro function spaces ${\operatorname{Ces}}_{p_2,q_2}(u_2,v_2)$ are characterized. In particular, two-sided…
It was pointed out to us that the proof of a crucial lemma (Lemma 5.3) in the paper is incorrect. Thus the approximation theorem (Theorem 0.1) for L^2 torsion of an amenable covering of a finite simplicial complex remains unproved. However,…
In this article, we consider the minimal $L^2$ integrals for the Hardy spaces and the Bergman spaces, and we present some relations between them, which can be regarded as the solutions of the finite points versions of Saitoh's conjecture…
This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…
We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings, The construction has been used in…
Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that in relevant cases such as…
We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_\tau : {B}_{p_1,q_1}^{s_1,\tau_1}(\Omega) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau_2}(\Omega)$ and $id_\tau : {F}_{p_1,q_1}^{s_1,\tau_1}(\Omega) \hookrightarrow…
In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d, \Omega\}$, in the subcritical…
The structure of non-compactness of optimal Sobolev embeddings of $m$-th order into the class of Lebesgue spaces and into that of all rearrangement-invariant function spaces is quantitatively studied. Sharp two-sided estimates of Bernstein…
In this paper we address smoothing-that is, optimisation-based-estimation techniques for localisation problems in the case where motion sensors are very accurate. Our mathematical analysis focuses on the difficult limit case where motion…
We show upper and lower embeddings of $\alpha_1$-modulation spaces in $\alpha_2$-modulation spaces for $0 \leq \alpha_1 \leq \alpha_2 \leq 1$, and prove partial results on the sharpness of the embeddings.
It is well-known that the embedding of the Sobolev space of weakly differentiable functions into H\"{o}lder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions…
The aim of this paper is to show that one can achieve convergence rates of $N^{-\alpha+ \delta}$ for $\alpha > 1/2$ (and for $\delta > 0$ arbitrarily small) for nonperiodic $\alpha$-smooth cosine series using lattice rules without random…
We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak H\"ormander condition. We prove continuous embeddings into Lorentz and intrinsic H\"older spaces. We also prove…
We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is…
The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions.…
We consider low-distortion embeddings for subspaces under \emph{entrywise nonlinear transformations}. In particular we seek embeddings that preserve the norm of all vectors in a space $S = \{y: y = f(x)\text{ for }x \in Z\}$, where $Z$ is a…
We study minimax density estimation on the product space $\mathbb{R}^{d_1}\times\mathbb{R}^{d_2}$. We consider $L^p$-risk for probability density functions defined over regularity spaces that allow for different level of smoothness in each…
We introduce truncated Besov and Triebel--Lizorkin function spaces and investigate their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis…