Bernstein-type Operators Research Articles

Convergence in distribution of the Bernstein–Durrmeyer kernel and pointwise convergence of a generalised operator for functions of bounded variation

We study the convergence of Bernstein type operators leading to two results. The first: The kernel Kn of the Bernstein–Durrmeyer operator at each point x∈(0,1) — that is Kn(x,t)dt — once standardised converges to the normal distribution. The second result computes the pointwise limit of a generalised Bernstein–Durrmeyer operator applied to — possibly discontinuous — functions f of bounded variation.

Bernstein–Jacobi-type operators preserving derivatives

A general frame for Bernstein-type operators that preserve derivatives is given. We introduce Bernstein-type operators based in the weighted classical Jacobi inner product on the interval [0, 1] that extend the well known Bernstein–Durrmeyer operator as well as some other types of Bernstein operators that appear in the literature. Apart from standard results, we deduce properties about the preservation of derivatives and prove that classical Jacobi orthogonal polynomials on [0, 1] are the eigenfunctions of these operators. We also study the limit cases when one of the parameters of the Jacobi polynomials is a negative integer. Finally, we study several numerical examples.

On the continuity in q of the family of the limit q-Durrmeyer operators

Abstract This study deals with the one-parameter family { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit q q -Durrmeyer operators. The continuity of this family with respect to the parameter q q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q ∈ [ 0 , 1 ] q\in \left[0,1] . When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 q=0 and fails at all q ∈ ( 0 , 1 ] . q\in \left(0,1]. In addition, a few estimates for the distance between two limit q q -Durrmeyer operators have been derived in the operator norm on C [ 0 , 1 ] C\left[0,1] .

On a Family of Parameter-Based Bernstein Type Operators with Shape-Preserving Properties

This article aims to introduce a new linear positive operator with a parameter. Our focus lies in analyzing the distinct characteristics and inherent properties exhibited by this operator. Additionally, we provide a proof of the convergence rate and present a revised version of the Voronovskaja theorem specifically tailored for this newly defined operator. Furthermore, we provide an upper bound for the error according to the modulus of continuity. Finally, the preservation of monotonicity and convexity by the operator is being investigated.

A new class of Bernstein-type operators obtained by iteration

"A new class of Bernstein-type operators are obtained by applying an iterative method of modi cations starting from the Bernstein operators. These operators have good properties of approximation of functions and of their derivatives."

Bernstein-Type Operators on the Unit Disk

We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyze the bivariate Bernstein–Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein–Stancu and the Bernstein-type operator on disk quadrants.

Bernstein-type operators on elliptic domain and their interpolation properties

Abstract The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) \left({{\mathcal{ {\mathcal B} }}}_{m}^{x}G)\left(x,z) and ( ℬ n z G ) ( x , z ) , \left({{\mathcal{ {\mathcal B} }}}_{n}^{z}G)\left(x,z), their products ( P m n G ) ( x , z ) \left({{\mathcal{P}}}_{mn}G)\left(x,z) , ( Q n m G ) ( x , z ) \left({{\mathcal{Q}}}_{nm}G)\left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) \left({{\mathcal{S}}}_{mn}G)\left(x,z) , ( T n m G ) ( x , z ) \left({{\mathcal{T}}}_{nm}G)\left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.

Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

Approximation by modified Bernstein polynomials based on real parameters

In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators.

On complex modified Bernstein-Stancu operators

<p style='text-indent:20px;'>The present paper deals with complex form of a generalization of perturbed Bernstein-type operators. Quantitative upper estimates for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation by these operators attached to functions analytic in a disk centered at the origin with radius greater than 1 are obtained in this study.</p>

Improve the approximation order of Bernstein type operators

In this study, we present a generalization of the well-known Bernstein operators based on an odd positive integer r denoted by K_(n,r) (f;x), first, we begin by studying the simultaneous approximation where we prove that the operator K_(n,r)^((s) ) (f;x) convergence to the function f^((s) ) (x) then we introduce and prove the Voronovskaja-type asymptotic formula when (r=3) giving us the order of approximation O(n^(-2) ) which is better than the order of the classical Bernstein operators O(n^(-1) ) followed by the error theorem and at the end, we give a numerical example to show the error of a test function and its first derivative taking different values of r.

On the shape-preserving properties of λ-Bernstein operators

We investigate the shape-preserving properties of λ-Bernstein operators B_{n,lambda } ( f;x ) that were recently introduced Bernstein-type operators defined by a new Beziér basis with shape parameter lambda in [ -1,1 ] . For this purpose, we express B_{n,lambda } ( f;x ) as a sum of a classical Bernstein operator and a sum of first order divided differences of f. Using this new representation, we prove that B_{n,lambda } ( f;x ) preserves monotonic functions for all lambda in [ -1,1 ] . However, we show by a counter example that B_{n,lambda } ( f;x ) does not preserve convex functions for some lambda in [ -1,1 ] . We present a weaker result for the case lambda in [ 0,1 ] for a special class of functions. Finally, we analyze the monotonicity of λ-Bernstein operators with n and show that B_{n,lambda } ( f;x ) is not monotonic with n for some λ if 1/2 <lambda leq 1.

On the Eigenstructure of the Modified Bernstein Operators

Starting from the well-known work of Cooper and Waldron published in 2000, the eigenstructure of various Bernstein-type operators has been investigated by many researchers. In this work, the eigenvalues and eigenvectors of the modified Bernstein operators Qn have been studied. These operators were introduced by S. N. Bernstein himself, in 1932, for the purpose of accelerating the approximation rate for smooth functions. Here, the explicit formulae for the eigenvalues and corresponding eigenpolynomials together with their limiting behavior are established. The results show that although some outcomes are similar to those for the Bernstein operators, there are essentially different ones as well.

An observer-based output tracking controller for electrically driven cooperative multiple manipulators with adaptive Bernstein-type approximator

AbstractThisarticle presents an observer-based output tracking control method for electrically actuated cooperative multiple manipulators using Bernstein-type operators as a universal approximator. This efficient mathematical tool represents lumped uncertainty, including external perturbations and unmodeled dynamics. Then, adaptive laws are derived through the stability analysis to tune the polynomial coefficients. It is confirmed that all the position and force tracking errors are uniformly ultimately bounded using the Lyapunov stability theorem. The theoretical achievements are validated by applying the proposed observer-based controller to a cooperative robotic system comprised of two manipulators transporting a rigid object. The outcomes of the introduced method are also compared to RBFNN, which is a powerful state-of-the-art approximator. The results demonstrate the efficacy of the introduced adaptive control approach in controlling the system even in the presence of disturbances and uncertainties.

Remarks on a Bernstein-type operator of Aldaz, Kounchev and Render

The Bernstein-type operator of Aldaz, Kounchev and Render (2009) is discussed. New direct results in terms of the classical second order modulus as well as in a modification following Marsden and Schoenberg are given.

On convergence of derivatives of Voronovskaja-type expansion for Bernstein type operators

In this paper, for new defined Bernstein type operators, we show that taking derivatives and taking the limit are sequence independent when the Voronovskaja-type expansion is differentiated.

Connections between the Approximation Orders of Positive Linear Operators and Their Max-Product Counterparts

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and Lp convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the Lp -norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.

Bivariate Bernstein polynomials that reproduce exponential functions

In this paper, we construct Bernstein type operators that reproduce exponential functions on simplex with one moved curved side. The operator interpolates the function at the corner points of the simplex. Used function sequence with parameters α and β not only are gained more modeling flexibility to operator but also satisfied to preserve some exponential functions. We examine the convergence properties of the new approximation processes. Later, we also state its shape preserving properties by considering classical convexity. Finally, a Voronovskaya-type theorem is given and our results are supported by graphics.

Better numerical approximation by λ-Durrmeyer-Bernstein type operators

The main object of this paper is to construct a new Durrmeyer variant of the ?-Bernstein type operators which have better features than the classical one. Some results concerning the rate of convergence in terms of the first and second moduli of continuity and asymptotic formulas of these operators are given. Moreover, we define a bivariate case of these operators and investigate the approximation degree by means of the total and partial modulus of continuity and the Peetre?s K-functional. A Voronovskaja type asymptotic and Gr?ss-Voronovskaja theorem for the bivariate operators is also proven. Further, we introduce the associated GBS (Generalized Boolean Sum) operators and determine the order of convergence with the aid of the mixed modulus of smoothness for the B?gel continuous and B?gel differentiable functions. Finally the theoretical results are analyzed by numerical examples.

On the rate of convergence of two generalized Bernstein type operators

In this paper, we introduce the Bezier variant of two new families of generalized Bernstein type operators. We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. By means of construction of suitable functions and the method of Bojanic and Cheng, we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.