Let $L\subset C$ be a regular Jordan curve. In this work, the approximation properties of the $p$-Faber-Laurent rational series expansions in the $\omega$ weighted Lebesgue spaces $L^p(L,\omega )$ are studied. Under some restrictive conditions upon the weight functions the degree of this approximation by a $k$th integral modulus of continuity in $L^p(L,\omega )$ spaces is estimated.

This article deals with the determination of the rate of convergence to the unit of each of three newly introduced perturbed normalized neural network operators of one hidden layer. These are given through the modulus of continuity of the function involved or its high order derivative that appears in the right-hand side of the associated Jackson type inequalities. The activation function is very general, in particular it can derive from any sigmoid or bell-shaped function. The right-hand sides of...

Using the $q$-Bernstein basis, we construct a new sequence $\left\{L_n\right\}$ of positive linear operators in $C[0,1].$ We study its approximation properties and the rate of convergence in terms of modulus of continuity.

The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators $${\widehat{M}}_{n,\alpha }(f,x)$$ for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators $${\widehat{M}}_{n,\alpha }(f,x)$$ for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators $${\widehat{M}}_{n,\alpha }(f,x)$$ for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators....

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.

We apply pluripotential theory to establish results in ${ℝ}^k$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $\Sigma \subset {ℝ}^k$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact sections includes...

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 $E$ un espace de Banach séparable de dimension infinie ; le sujet de cette étude est l’approximation de fonctions de classe $C^k$ définies sur un ouvert $\Omega$ de $E$ à valeurs dans un espace de Banach $F$ par des fonctions de classe $C^{\infty }$. Le principal résultat est : si $E$ est un espace de Hilbert, l’ensemble des applications de classe $C^{\infty }$ de $\Omega$ dans $F$ est dense dans l’ensemble des applications de classe $C^{2k-1}$ muni de la $C^k$ topologie fine. Comme corollaire, on montre que l’étude des variétés hilbertiennes de classe $C^k$...