Abstract

In the present paper, we propose a Baskakov operator of integral type using a function φ on [0,∞) with the properties: φ(0)=0,φ′>0 on [0,∞) and limx→∞φ(x)=∞. The proposed operators reproduce the function φ and constant functions. For the constructed operator, some approximation properties are studied. Voronovskaja asymptotic type formulas for the proposed operator and its derivative are also considered. In the last section, the interest is focused on weighted approximation properties, and a weighted convergence theorem of Korovkin’s type on unbounded intervals is obtained. The results can be extended on the interval (−∞,0] (the symmetric of the interval [0,∞) from the origin).

Highlights

  • Symmetry 2021, 13, 1747. https://Since 1912, when Bernstein introduced his famous polynomials, in order to proveWeierstrass’s fundamental theorem in the most elegant form, the field of approximation theory proved its usefulness many times with various applications.Different kinds of generalizations of Bernstein operators have become a powerful tool in solving differential equations and highlight their applicability in domains such as numerical analysis, computer aided geometric design or artificial neural networks

  • We introduced and discussed the approximation properties of a new class of Baskakov operators of integral type, defined on right-unbounded interval [0, ∞)

  • Our first important result is a Voronovskaja type theorem for our operators Gn defined in (2)

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Summary

Introduction

Weierstrass’s fundamental theorem in the most elegant form, the field of approximation theory proved its usefulness many times with various applications. Since 1950, the concept of linear and positive operators has gained great importance: the uniform convergence of a sequence of operators to a continuous function on a finite and closed interval can be proved. This theorem is known as the famous Popoviciu–. We introduced and discussed the approximation properties of a new class of Baskakov operators of integral type, defined on right-unbounded interval [0, ∞). Our purpose is to study the local approximation properties of the proposed operators of integral type. We study the weighted approximation properties, implying the weighted modulus of continuity introduced by Ispir and Atakut in [6]

Convergence of the Gn Operators
Weighted Approximation

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