Generalized Bernstein Operators Research Articles

Voronovskaja type theorem for some nonpositive Kantorovich type operators

In this paper we will study a Voronovskaja type theorem and a simultaneous approximation result for a new class of generalized Bernstein operators. The new operators are obtained using a generalization of Kantorovich's method, namely, we will introduce a sequence of operators $K_n^l=D^l\circ B_{n+l}\circ I^l$, where $B_{n+l}$ are Bernstein operators, $D^{l}f=f^{(l)}+a_{l-1}f^{(n-1)}+\dots+a_1f'+a_0f$ is a differential operator with constant coefficients $a_j,\ j\in\{0,\dots,l-1\}$ and $I^{l}$ a corresponding antiderivative operator such that $D^{l}\circ I^{l}=Id$.

Statistical convergence of bivariate generalized bernstein operators via four-dimensional infinite matrices

Our main aim in this work is to construct an original extension of bivariate Bernstein type operators based on multiple shape parameters to give an application of four-dimensional infinite matrices to approximation theory, and prove some Korovkin theorems using two summability methods: a statistical convergence method which is stronger than the classical case and a power series method. We obtain the rate of generalized statistical convergence, and the rate of convergence for the power series method. Moreover, we provide some computer graphics to numerically analyze the efficiency and accuracy of convergence of our operators and obtain corresponding error plots. All the results that have been obtained in the present paper can be extended to the case of n-variate functions.

On a new family of generalized Bernstein operators

"In this paper we remark that alfa-Bernstein operators, introduced by X. Y. Chen et al., are combinations of two known operators (Stancu and Bernstein operators) and we establish the preservation of global smoothness properties by these linear operators, the global smoothness being expressed by a Lipschitz condition with a certain second order modulus of continuity."

Some numerical applications of generalized Bernstein operators

In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.

Generalized Bernstein–Durrmeyer operators of blending type

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter $$\rho $$ . We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

A Kantorovich variant of a generalized Bernstein operators

A Kantorovich variant of a generalized Bernstein operators

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on polynomial spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that $\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several examples, we exhibit the diversity of behaviours in this more general setting. We also prove that for sufficiently large dimensions, there always exist generalized Bernstein operators fixing $\mathbf{1}$ and $f_1$, and converging to the identity.

Shape-preserving properties of a new family of generalized Bernstein operators

In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called (alpha,q)-Bernstein operators, denoted by T_{n,q,alpha}(f). We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of T_{n,q,alpha}(f) to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to f(x). We also obtain the monotonicity with n and q of T_{n,q,alpha}(f).

Degree of Approximation for Bivariate Generalized Bernstein Type Operators

In this paper we study an extension of the bivariate generalized Bernstein operators based on a non-negative real parameters. For these operators we obtain the order of approximation using Peetre’s K-functional, a Voronovskaja type theorem and the degree of approximation by means of the Lipschitz class. Further, we consider the Generalized Boolean Sum operators of generalized Bernstein type and we study the degree of approximation in terms of the mixed modulus of continuity. Finally, we show the comparisons by some illustrative graphics in Maple for the convergence of the operators to certain functions.

Approximation of functions by a new family of generalized Bernstein operators

The main object of this paper is to construct a new generalization of the Bernstein operator, depending on a non-negative real parameter. We investigate some elementary properties of this operator, such as end point interpolation, linearity and positivity, etc. By using these generating operators, we provide another proof of the Weierstrass Approximation Theorem. We give the detailed proofs to the rate of convergence and Voronovskaja type asymptotic estimate formula for the operators. Moreover, an upper bound for the error is obtained in terms of the usual modulus of continuity. Shape preserving properties of the generalized Bernstein operators are also studied. It is proved that monotonic or convex functions produce monotonic or convex generalized Bernstein polynomials.

The generalization of the Bernstein operator on any finite interval

Abstract The paper presents the generalized form of the Bernstein operator associated with any real-valued function f : [ a , b ] → ℝ {f\colon[a,b]\to\mathbb{R}} . For this generalized Bernstein operator, we study the qualitative and quantitative aspects concerning uniform convergence, order of approximation and asymptotic behavior.

Bivariate generalized Bernstein operators and their application to Fredholm integral equations

We introduce and study the sequence of bivariate Generalized Bernstein operators {Bm,s}m,s, m, s ? N, Bm,s=I?(I?Bm)s, Bi m = Bm(Bi?1 m), where Bm is the bivariate Bernstein operator. These operators generalize the ones introduced and studied independently in the univariate case by Mastroianni and Occorsio [Rend. Accad. Sci. Fis. Mat. Napoli 44 (4) (1977), 151- 169] and by Micchelli [J. Approx. Theory 8 (1973), 1-18] (see also Felbecker [Manuscripta Math. 29 (1979), 229-246]). As well as in the one-dimesional case, for m fixed the sequence {Bm,s(f)}s can be successfully employed in order to approximate ?very smooth? functions f by reusing the same data points f (i/m,j/m), i=0,1,...,m, j=0,1,...,m, since the rate of convergence improves as s increases. A stable and convergent cubature rule on the square [0,1]2, based on the polynomials Bm,s(f) is constructed. Moreover, a Nystrom method based on the above mentioned cubature rule is proposed for the numerical solution of two-dimensional Fredholm integral equations on [0, 1]2. The method is numerically stable, convergent and the involved linear systems are well conditioned. Some algorithm details are given in order to compute the entries of the linear systems with a reduced time complexity. Moreover the procedure can be significantly simplified in the case of equations having centrosymmetric kernels. Finally, some numerical examples are provided in order to illustrate the accuracy of the cubature formula and the computational efficiency of the Nystrom method.

Weighted Approximation by Analogues of Bernstein Operators for Rational Functions

Weighted modifications of generalized Bernstein operators in rational functions (Videnskii operators) are introduced. Their convergence in weighted spaces is studied.

Approximation of Generalized Bernstein Operators

Approximation of Generalized Bernstein Operators

Generalized Bézier curves and surfaces based on Lupaş [formula omitted]-analogue of Bernstein operator

In this paper, a new generalization of Bézier curves with one shape parameter is constructed. It is based on the Lupaş q-analogue of Bernstein operator, which is the first generalized Bernstein operator based on the q-calculus. The new curves have some properties similar to classical Bézier curves. Moreover, we establish degree evaluation and de Casteljau algorithms for the generalization. Furthermore, we construct the corresponding tensor product surfaces over the rectangular domain, and study the properties of the surfaces, as well as the degree evaluation and de Casteljau algorithms. Compared with q-Bézier curves and surfaces based on Phillips q-Bernstein polynomials, our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces. The shape parameters provide more convenience for the curve and surface modeling.

Weighted approximation by analogues of Bernstein operators for rational functions

Weighted modifications of generalized Bernstein operators in rational functions (Videnskii operators) are introduced. Their convergence in weighted spaces is studied.

Bernstein type operators having 1 and x j as fixed points

AbstractFor certain generalized Bernstein operators {L n} we show that there exist no i, j ∈ {1, 2, 3,…}, i < j, such that the functions e i(x) = x i and e j (x) = x j are preserved by L n for each n = 1, 2,… But there exist infinitely many e i such that e 0(x) = 1 and e j (x) = x j are its fixed points.

Simultaneous approximation on generalized Bernstein–Durrmeyer operators

In the present paper, we study some theorems on approximation of the r-th derivative of a given function f by corresponding r-th derivative of the generalized Bernstein operator.

Optimality of generalized Bernstein operators

We show that a certain optimality property of the classical Bernstein operator also holds, when suitably reinterpreted, for generalized Bernstein operators on extended Chebyshev systems.

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0, and a second function f 1 such that f 1/f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C[a, b] → U n with strictly increasing nodes, fixing $${f_{0}, f_{1} \in U_{n}}$$ . If $${U_{n} \subset U_{n + 1}}$$ and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1. In particular, if f 0, f 1, . . . , f n is a basis of U n such that the linear span of f 0, . . . , f k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1. The second main result says that under the above assumptions the following inequalities hold $$B_{n} f \geq B_{n+1} f \geq f$$ for all (f 0, f 1)-convex functions $${f \in C \, [ a, b]}$$ . Furthermore, B n f is (f 0, f 1)-convex for all (f 0, f 1)-convex functions $${f \in C \, [a, b]}$$ .