The complete asymptotic expansion for Bernstein operators

In this paper we study the asymptotic behavior of the classical Bernstein operators, applied to q-times continuously differentiable functions. Our main results extend the results of S.N. Bernstein and R.G. Mamedov for all q-odd natural numbers and thus generalize the theorem of E.V. Voronovskaja. The exact degree of approximation is also proved.     

Approximation by Algebraic Polynomials on Rectangles

A direct theorem for approximation by algebraic polynomials in two variables with different degrees in each variable in L_p-metric (1 p ) on rectangles is proved, and the dependence of the constants on various parameters is studied.     

The Rate of Approximation by Rational Bernstein Functions in Terms of Second Order Moduli of Continuity

The aim of this note is to study the impact of the weight numbers in two concrete cases on the rate of approximation by rational Bernstein functions. The approximation is measured in terms of second order moduli of continuity. We consider also the case of rational Bernstein curves and the role of the weights to modify the shape of the curves.     

New estimates in Voronovskaja’s theorem

In the present article we establish pointwise variant of E. V. Voronovskaja’s 1932 result, concerning the degree of approximation of Bernstein operator, applied to functions f ∈ C ³[0, 1].     

On the mechanism of stomatocyte-echinocyte transformations of red blood cells: experiment and theoretical model

This study represents an attempt to achieve a better understanding of the stomatocyte-echinocyte transition in the shape of red blood cells. We determined experimentally the index of cell shape at various ionic strengths and osmolarities for native and trypsin-treated human erythrocytes. For every given composition of the outer phase, we calculated the ionic strength in the cells and the transmembrane electric potential using a known theoretical model. Next, we described theoretically the electric double layers formed on both sides of the cell membrane, and derived expressions for the tensions of the two membrane leaflets. Taking into account that the cell-shape index depends on the tension difference between the two leaflets, we fitted the experimental data with the constructed physicochemical model. The model, which agrees well with the experiment, indicates that the tension difference between the two leaflets is governed by the different adsorptions of counterions at the two membrane surfaces, rather than by the direct contribution of the electric double layers to the membrane tension. Thus, with the rise of the ionic strength, the counterion adsorption increases stronger at the outer leaflet, whose stretching surface pressure becomes greater, and whose area expands relative to that of the inner leaflet. Hence, there is no contradiction between the bilayer-couple hypothesis and the electric double layer theory, if the latter is upgraded to account for the effect of counterion-adsorption on the membrane tension. The developed quantitative model can be applied to predict the shape index of cells upon a stomatocyte-discocyte-echinocyte transformation at varying composition of the outer medium.     

Piecewise Linear Interpolation With Nonequidistant Nodes

Pointwise and norm estimates for approximation of continuous function by piecewise linear interpolant with nonequidistant nodes are proved. Special attention is paid to the value of the absolute constants in front of the modulus.     

Quantitative Voronovskaya-Type Results for Polynomially Bounded Functions

We prove quantitative Voronovskaya-type results and Grüss–Voronovskaya inequalities for polynomially bounded functions. The results are given in terms of suitable K-functionals and applied to linking Szász–Mirakyan and Baskakov operators.     

Approximation by Kantorovich-Bernstein polynomials inL p(0<p<1)-metric

The purpose of the present paper is to evaluate the error of the approximation of the function f∈L₁[0,1] by Kantorovich-Bernstein polynomials in L_p-metric (0<p<1).     

Approximation of a Continuous Curve by its Bernstein-B��zier Operator

In this paper we consider integer and rational parametric Bézier curves and study the distance between the curve and its control polygon. To measure the distance we use first and second order moduli of smoothness of vector-valued function. We consider also NURBS curves with equidistant knots. Some direct approximation theorems will be presented.