Bernstein Kantorovich Research Articles

Kantorovich Variant of the Blending Type Bernstein Operators

In this paper, we introduce a novel class of blending-type Bernstein–Kantorovich operators. These operators depend on three parameters: α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, and s. We establish results on the uniform convergence and rate of convergence of these operators in terms of the first and second order modulus of continuity. We also investigate the shape-preserving properties of the operators, such as monotonicity and convexity, for each choice of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, and s. Finally, we provide graphical and numerical results to illustrate the accuracy of the operators and to demonstrate how they approach certain functions.

ψ$$ \psi $$‐Bernstein–Kantorovich operators

In this article, we introduce a modified class of Bernstein–Kantorovich operators depending on an integrable function and investigate their approximation properties. By choosing an appropriate function , the order of approximation of our operators to a function is at least as good as the classical Bernstein–Kantorovich operators on the interval . We compared the operators defined in this study not only with Bernstein–Kantorovich operators but also with some other Bernstein–Kantorovich type operators. In this paper, we also study the results on the uniform convergence and rate of convergence of these operators in terms of the first‐ and second‐order moduli of continuity, and we prove that our operators have shape‐preserving properties. Finally, some numerical examples which support the results obtained in this paper are provided.

Approximation by a new kind of $$(\lambda ,\mu )$$-Bernstein–Kantorovich operators

Approximation by a new kind of $$(\lambda ,\mu )$$-Bernstein–Kantorovich operators

Bernstein–Kantorovich operators, approximation and shape preserving properties

Inspired by the construction of Bernstein and Kantorovich operators, we introduce a family of positive linear operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{K}_n$$\\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.

Approximation by bivariate Bernstein–Kantorovich–Stancu operators that reproduce exponential functions

In this study, we construct a Stancu-type generalization of bivariate Bernstein–Kantorovich operators that reproduce exponential functions. Then, we investigate some approximation results for these operators. We use test functions to prove a Korovkin-type convergence theorem. Then, we show the rate of convergence by the modulus of continuity and give a Voronovskaya-type theorem. We give a covergence comparison about bivariate Bernstein–Kantorovich–Stancu operators and their exponential form.

Approximation Theorems for Complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich Operators

In this paper, we introduce the complex form of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operators. Respectively, upper quantitative estimates for the complex α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-Bernstein–Kantorovich operator and its derivatives, Voronovskaya type result and the exact order of approximation of these operators are studied.

Different Variants of Bernstein Kantorovich Operators and Their Applications in Sciences and Engineering Field

In this article, we investigate various Bernstein-Kantorovich variants together with their approximation properties. Nowadays, these variants of Bernstein-Kantorovich operators have been a source of inspiration for researchers as it helps to approximate integral functions also which is not feasible in the case of discrete operators. Chaos theory has also been referred to as complexity theory. Using chaos theory complexity is also reduced as in approximation theory. Thus in order to reduce complexity and to have better understanding of images in sciences and engineering field, sampling Kantorovich operators of approximation theory are widely used in this regard for enhancement of images. Thus, we discuss the important applications of Kantorovich operators depicting pragmatic and theoretical aspects of approximation theory.

Some elementary properties of Bernstein–Kantorovich operators on mixed norm Lebesgue spaces and their implications in fractal approximation

On the one hand, the notion of mixed norm spaces has attracted considerable attention in fields such as harmonic analysis and PDE. On the other hand, a particular modification of the Bernstein operator, the so-called Bernstein–Kantorovich operator, has been of special interest for the approximation of the classical Lp-functions. This note has a double purpose. First, we record some elementary approximation properties of the Bernstein–Kantorovich operators on the mixed norm Lebesgue spaces. In the second part, we construct self-referential (fractal) counterparts to the functions belonging to the mixed norm Lebesgue spaces and introduce fractal operators on these spaces. With the help of the Bernstein–Kantorovich operators, we obtain a fractal approximation process on the mixed norm Lebesgue spaces. Furthermore, using the multivariate Haar system, we provide a Schauder basis consisting of self-referential functions for the mixed norm Lebesgue spaces, which we call the Bernstein–Kantorovich fractal Haar system.

Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein–Kantorovich operators

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative.

Bivariate modified Bernstein–Kantorovich operators for the numerical solution of two‐dimensional fractional Volterra integral equations

A class of two‐dimensional fractional Volterra integral equations (2D‐FVIEs) of the second kind is considered. The solution may have unbounded derivatives near the integral domain boundary. Therefore, smoothing transformations are employed to change the original 2D‐FVIEs into new transformed 2D‐FVIEs with better regularity. The novelty in this research concerns both the theoretical investigation of the bivariate modified Bernstein–Kantorovich (B‐MBK) operators and the numerical application of these operators for approximating the unknown solution of 2D‐FVIEs. In this regard, an algorithm is given utilizing the B‐MBK operators and discretization that approximates the solution of the transformed discretized equation. Further, an inverse transformation is applied to obtain the solution of the original equation. Additionally, we illustrate the applicability of the proposed method on examples from the literature.

Approximation properties of univariate and bivariate new class $$\lambda $$-Bernstein–Kantorovich operators and its associated GBS operators

Approximation properties of univariate and bivariate new class $$\lambda $$-Bernstein–Kantorovich operators and its associated GBS operators

A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator.

With the development of intelligent manufacturing and mechatronics, robotic manipulators are used more widely. There are complex noises and external disturbances in many application cases that affect the control accuracy of the manipulator servo system. On the basis of previous research, this paper improves the manipulator controller, introduces the Bernstein-Kantorovich-Stancu (BKS) operator, and proposes a modified robotic manipulator controller to improve the error tracking accuracy of the manipulator controller when observing complex disturbances and noises. In addition, in order to solve the problem that the coupling between the external disturbances of each axis of the manipulator leads to a large amount of computation when observing disturbances, an improved full-order observer is designed, which simplifies the parameters of the controller combined with the BKS operator and reduces the complexity of the algorithm. Through a theoretical analysis and a simulation test, it was verified that the proposed manipulator controller could effectively suppress external disturbance and noise, and the application of the BKS operator in the manipulator servo system control is feasible and effective.

Hybrid Operators for Approximating Nonsmooth Functions and Applications on Volterra Integral Equations with Weakly Singular Kernels

This research concerns some theoretical and numerical aspects of hybrid positive linear operators for approximating continuous functions on that have unbounded derivatives at the initial point. These operators are defined by using Modified Bernstein–Kantorovich operators where n is positive integer, is a fixed constant and reduces to the classical Bernstein–Kantorivich operators when To show the importance and the applicability of the given hybrid operators we develop an algorithm which implements them for solving the second kind linear Volterra integral equations with weakly singular kernels. Furthermore, applications are also performed on first kind integral equations, by utilizing regularization. Eventually, it is shown that the numerical realization of the given algorithm is easy and computationally efficient and gives accurate approximations to nonsmooth solutions.

Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials in computational disciplines, we propose a new generalization of Bernstein–Kantorovich operators involving shape parameters λ, α and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program.

Generalized Bernstein Kantorovich operators: Voronovskaya type results, convergence in variation

This paper includes Voronovskaya type results and convergence in variation for the exponential Bernstein Kantorovich operators. The Voronovskaya type result is accompanied by a relation between the mentioned operators and suitable auxiliary discrete operators. Convergence of the operators with respect to the variation seminorm is obtained in the space of functions with bounded variation. We propose a general framework covering the results provided by previous literature.

The family of Bernstein–Kantorovich operators with shifted knots

The family of Bernstein–Kantorovich operators with shifted knots

Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

Korovkin type theorem for Bernstein–Kantorovich operators via power summability method

In this paper we will prove the Korovkin type theorem for Bernstein–Kantorovich type operators via $$A-$$ statistical convergence and power summability method. Also we give the rate of the convergence related to the above summability methods and in the last sections we give a kind of Voronovskaya type theorem for $$A-$$ statistical convergence and Gruss–Voronovskaya type theorem.

Semi-discrete Quantitative Voronovskaya-Type Theorems for Positive Linear Operators

Because all the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor’s formula which is a very particular case of Lagrange–Hermite interpolation formula, in this paper we obtain semi-discrete quantitative Voronovskaya-type theorems based on various other Lagrange–Hermite interpolation formulas. Applications to Bernstein–Kantorovich and to Bernstein operators are presented.

Approximation Properties of Generalized $$\lambda$$-Bernstein–Kantorovich Type Operators

Approximation Properties of Generalized $$\lambda$$-Bernstein–Kantorovich Type Operators