UDC

517.518.8

Authors

Polovinkina Yuliya Stanislavovna
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
e-mail: u.polovinkina@narfu.ru

Abstract

It is known that for the Bernstein polynomials and some of their classical generalizations belonging to the class of linear positive operators the order of approximation of these operators is not improved in the development of smoothness. The presence of the derivative of order above two ceases to affect on the degree of convergence of the Bernstein polynomials to the generating function. The Bernstein polynomials possess a remarkable property of simultaneous approximation to function and its derivatives. This fact makes them a convenient tool in the design of various numerical models (e.g., for the giving data approximation in computational algorithms). There are several approaches to the consecutive orders of polynomial operators, which would solve a problem of approximation rate of continuously differentiable functions. Most often the question is in the definition of some modifications of the initial polynomials, such as the consecutive orders of the Bernstein type, Kirov modifications. The paper proposes a fundamentally different method of generalization of classical polynomials, which allows maintaining their linearity and positivity, and the methods of proving theorems based on it. At the same time this method leads to the operators production, responding to the smoothness of the function. First the iterative splines are designed on the basis of the Bernstein polynomials. They have a higher degree of convergence to the generating function than the original operators. The relevant theorems of approximation of continuous and differentiable functions and the moment about mean estimation are presented. It is shown that the designed splines have the same disadvantage as the generator polynomials. The spline fitting with the derivative of order above two does not improve. We study the spline modifications with the essentially increasing order of convergence to the generating function in the development of smoothness. The paper studies the essential approximating properties of the obtained sequences of operators, proves the corresponding theorems of the Popovichiu and Voronovskaya–Bernstein theorem types.

Keywords

iterative spline, Bernstein polynomials, modifications

References

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