Starting from the study of the Shepard nonlinear operator of max-prod type in (Bede, Nobuhara et al., 2006, 2008), in the book (Gal, 2008), Open Problem 5.5.4, pp. 324–326, the Bleimann-Butzer-Hahn max-prod type operator is introduced and the question of the approximation order by this operator is raised. In this paper firstly we obtain an upper estimate of the approximation error of the form ${\omega }_1(f;{(1+x)}^{\frac{3}{2}}\sqrt{x/n})$. A consequence of this result is that for each compact subinterval $[0,a]$, with arbitrary $a>0$, the order of uniform...

We define a new kind of Durrmeyer-type summation-integral operators and study a global direct theorem for these operators in terms of the Ditzian-Totik modulus of smoothness.

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...

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....

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 $H(K,E)$.

The direct and inverse problems of approximation theory in the subspace of weighted generalized grand Lebesgue spaces of 2π-periodic functions with the weights satisfying Muckenhoupt's condition are investigated. Appropriate direct and inverse theorems are proved. As a corollary some results on constructive characterization problems in generalized Lipschitz classes are presented.

It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on $ℝ=(-\infty ;+\infty )$.

Let $u$ be harmonic in a bounded domain $D$ with smooth boundary. We prove that if the boundary values of $u$ belong to $L^p\left(\sigma \right)$, where $p\ge 2$ and $\sigma$ denotes the surface measure of $\partial D$, then it is possible to approximate $u$ uniformly by function of bounded variation. An example is given that shows that this result does not extend to $p\&lt;2$.

We establish direct estimates for the q-Baskakov operator introduced by Aral and Gupta in [2], using the second order Ditzian-Totik modulus of smoothness. Furthermore, we define and study the limit q-Baskakov operator.