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.

General Besov and Triebel-Lizorkin spaces on domains with irregular boundary are compared with the completion, in those spaces, of the subset of infinitely continuously differentiable functions with compact support in the same domains. It turns out that the set of parameters for which those spaces coincide is strongly related to the fractal dimension of the boundary of the 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 investigate the approximation properties of the partial sums of the Fourier series and prove some direct and inverse theorems for approximation by polynomials in weighted Orlicz spaces. In particular we obtain a constructive characterization of the generalized Lipschitz classes in these spaces.

Soit $K$ un compact de $C^n$ de la forme $K={\Pi }_{i=1}^r{K}_i$ où chaque $K_i$ est soit l’adhérence d’un domaine strictement pseudoconvexe dans $C^{n_i}$, soit l’adhérence d’un polyèdre de Weil régulier, ou encore un compact de $C$. $E$ étant un espace de Fréchet, on montre que lorsque $f$ appartient à ${\mathbf{C}}^1(K,E)$ avec $\bar{\partial }f\cong 0$ alors $f$ est approchable uniformément sur $K$ par des fonctions holomorphes au voisinage de $K$ et à valeurs dans $E$. On donne également des résultats de localisation pour l’espace $H(K,E)$.

Soit $X$ un espace de Banach complexe, et notons $B\left(R\right)\subset X$ la boule de rayon $R$ centrée en $0$. On considère le problème d’approximation suivant: étant donnés $0\<r\<R$, $ϵ\>0$ et une fonction $f$ holomorphe dans $B\left(R\right)$, existe-t-il toujours une fonction $g$, holomorphe dans $X$, telle que $|f-g|\<ϵ$ sur $B\left(r\right)$? On démontre que c’est bien le cas si $X$ est l’espace $l^1$ des suites sommables.

Let $X$ be a complex Banach space. Recall that $X$ admits afinite-dimensional Schauder decompositionif there exists a sequence ${\left\{X_n\right\}}_{n=1}^{\infty }$ of finite-dimensional subspaces of $X,$ such that every $x\in X$ has a unique representation of the form $x={\sum }_{n=1}^{\infty }{x}_n,$ with $x_n\in X_n$ for every $n.$ The finite-dimensional Schauder decomposition is said to beunconditionalif, for every $x\in X,$ the series $x={\sum }_{n=1}^{\infty }{x}_n,$ which represents $x,$ converges unconditionally, that is, ${\sum }_{n=1}^{\infty }{x}_{\pi \left(n\right)}$ converges for every permutation $\pi$ of the integers. For short, we say that $X$ admits an unconditional F.D.D.We...