The aim of the paper is twofold. First we give a survey of some recent results concerning the asymptotic behavior of the entropy and approximation numbers of compact Sobolev embeddings. Second we prove new estimates of approximation numbers of embeddings of weighted Besov spaces in the so called limiting case.

Upper estimates are obtained for approximation and entropy numbers of the embeddings of weighted Sobolev spaces into appropriate weighted Orlicz spaces. Results are given when the underlying space domain is bounded and for certain unbounded domains.

Let G be a locally compact Hausdorff group with Haar measure, and let L⁰(G) be the space of extended real-valued measurable functions on G, finite a.e. Let ϱ and η be modulars on L⁰(G). The error of approximation ϱ(a(Tf - f)) of a function $f\in {\left(L⁰\left(G\right)\right)}_{\varrho +\eta }\cap DomT$ is estimated, where $\left(Tf\right)\left(s\right)={\int }_{G}K(t-s,f\left(t\right))dt$ and K satisfies a generalized Lipschitz condition with respect to the second variable.

We show that in a super-reflexive Banach space, the conditionality constants $k_N\left(ℬ\right)$ of a quasi-greedy basis ℬ grow at most like $O\left({\left(logN\right)}^{1-\epsilon }\right)$ for some 0 < ε < 1. This extends results by the third-named author and Wojtaszczyk (2014), where this property was shown for quasi-greedy bases in $L_p$ for 1 < p < ∞. We also give an example of a quasi-greedy basis ℬ in a reflexive Banach space with $k_N\left(ℬ\right)\approx logN$.

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy...

Let id be the natural embedding of the Sobolev space $W_p^l\left(\Omega \right)$ in the Zygmund space $L_q{\left(logL\right)}_a\left(\Omega \right)$, where $\Omega ={(0,1)}^n$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers $e_k\left(id\right)$ of this embedding and show that $e_k\left(id\right)\asymp k^{-\eta }$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.

Our purpose in this paper is to provide a general approach to model selection via penalization for Gaussian regression and to develop our point of view about this subject. The advantage and importance of model selection come from the fact that it provides a suitable approach to many different types of problems, starting from model selection per se (among a family of parametric models, which one is more suitable for the data at hand), which includes for instance variable selection in regression models,...

We prove asymptotic formulas for the behavior of Gelfand and Kolmogorov numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces of radial distributions. Our method works also for Weyl numbers.

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1/2}cₙ\left(cov\left(K\right)\right)\le c_K(1+{\sum }_{k=1}^{ⁿ}{k}^{-1/2}{e}_k\left(K\right))$, n ∈ ℕ, and $k^{1/2}{c}_{k+n}\left(cov\left(K\right)\right)\le c[lo{g}^{1/2}(n+1)\epsilon ₙ\left(K\right)+{\sum }_{j=n+1}^{\infty }{\epsilon }_j\left(K\right)/\left(jlo{g}^{1/2}(j+1)\right)]$, k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ${\epsilon }_k\left(K\right)$ and $e_k\left(K\right)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly...