Boolean Sum Operators Research Articles

Quantitative estimates for the tensor product (p,q)-Balázs-Szabados operators and associated generalized Boolean sum operators

In this study, we give some approximation results for the tensor product of (p,q)-Bal?zs-Szabados operators associated generalized Boolean sum (GBS) operators. Firstly, we introduce tensor product (p,q)-Bal?zs-Szabados operators and give an uniform convergence theorem of these operators on compact rectangular regions with an illustrative example. Then we estimate the approximation for the tensor product (p,q)-Bal?zs-Szabados operators in terms of the complete modulus of continuity, the partial modulus of continuity, Lipschitz functions and Petree?s K-functional corresponding to the second modulus of continuity. After that, we introduce the GBS operators associated the tensor product (p,q)-Bal?zs-Szabados operators. Finally, we improve the rate of smoothness by the mixed modulus of smoothness and Lipschitz class of B?gel continuous functions for the GBS operators.

Extension of Some Generalized Hermite-Type Interpolation Operators to the Triangle with One Curved Side

We construct some generalized Hermite-type interpolation operators for the case of a triangle with one curved side, and we consider their product and Boolean sum operators. We study the interpolation properties of these operators, the orders of accuracy and the remainders of the interpolation formulas. Finally, we give some numerical examples.

Interpolation Operators on a Triangle With all Curved Sides

AbstractThis paper contains a survey regarding interpolation and Cheney-Sharma type operators defined on a triangle with all curved sides; we considers as well some of the product and Boolean sum operators. We study their interpolation properties and the degree of exactness

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.

Blending Type Approximation by GBS Operators of Generalized Bernstein–Durrmeyer Type

In this article we study an extension of the bivariate generalized Bernstein–Durrmeyer operators based on a non-negative real parameter. For these operators we get a Voronovskaja type theorem, the order of approximation using Peetre’s K-functional and the degree of approximation by means of the Lipschitz class. Further, we introduce the generalized boolean sum operators of generalized Bernstein–Durrmeyer type and we estimate the degree of approximation in terms of the mixed modulus of smoothness.

Extension of some particular interpolation operators to a triangle with one curved side

We extend some Nielson and Marshall type interpolation operators to the case of a triangle with one curved side. We study the interpolation properties of the obtained operators and of their product and Boolean sum operators, the orders of accuracy and the remainders of the corresponding interpolation formulas. Finally, we give some numerical examples.

Kantorovich variant of a new kind of q‐Bernstein–Schurer operators

Ren and Zeng (2013) introduced a new kind of q‐Bernstein–Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q‐Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K‐functional. Next, we introduce the bivariate case of q‐Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K‐functional. Finally, we define the generalized Boolean sum operators of the q‐Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

Blending type approximation by q‐generalized Boolean sum of Durrmeyer type

This paper is in continuation of the work performed by Kajla et al. (Applied Mathematics and Computation 2016; 275 : 372–385.) wherein the authors introduced a bivariate extension of q‐Bernstein–Schurer–Durrmeyer operators and studied the rate of convergence with the aid of the Lipschitz class function and the modulus of continuity. Here, we estimate the rate of convergence of these operators by means of Peetre's K‐functional. Then, the associated generalized Boolean sum operator of the q‐Bernstein–Schurer–Durrmeyer type is defined and discussed. The smoothness properties of these operators are improved with the help of mixed K‐functional. Furthermore, we show the convergence of the bivariate Durrmeyer‐type operators and the associated generalized Boolean sum operators to certain functions by illustrative graphics using Maple algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

Extension of Some Positive and Linear Operators to Domains with Curved Sides

There are constructed some Cheney-Sharma type operators defined on a square with one curved side. They are extensions of the Cheney-Sharma type operators of second kind, given by E.W. Cheney and A. Sharma in [14], to the case of a curved sided domain. There are constructed the univariate Cheney-Sharma type operators, their product and boolean sum operators and there are studied their properties, their orders of accuracy and the remainders of the corresponding approximation formulas. Finally, there are given some illustrative examples.

GBS Operators of Lupaş–Durrmeyer Type Based on Polya Distribution

In this paper, we study an extension of the bivariate Lupas–Durrmeyer operators based on Polya distribution. For these operators we get a Voronovskaja type theorem and the order of approximation using Peetre’s K-functional. Then, we construct the Generalized Boolean Sum operators of Lupas–Durrmeyer type and estimate the degree of approximation in terms of the mixed modulus of smoothness.

Interpolation Operators on Some Triangles with Curved Sides

Abstract This paper contains a survey regarding interpolation and Bernstein-type operators defined on triangles having one or all curved sides; we consider as well some of the product and Boolean sum operators. We study the interpolation properties, the orders of accuracy and the remainders of the generated approximation formulas.

Interpolation operators on a triangle with two and three curved edges

We construct Lagrange, Hermite and Birkhoff-type operators, which interpolate a given function and some of its derivatives on the border of a triangle with two and three curved edges. We also consider some of their product and boolean sum operators. We study the interpolation properties and the degree of exactness of the constructed operators.

About approximation of B-continuous functions of several variables by generalized Boolean sum operators of Bernstein type on a simplex

The aim of this paper is to study the convergence of the sequence of generalized Boolean sum (GBS) operators (UBm)m≥1 for B-continuous functions f ∈ Cb(∆4).

Interpolation operators on a triangle with one curved side

We construct certain Lagrange, Hermite and Birkhoff-type operators, which interpolate a given function and some of its derivatives on the border of a triangle with one curved side, as well as some of their product and Boolean sum operators. We study the interpolation properties and the order of accuracy (degree of exactness and precision set) of the constructed operators, respectively the remainders of the corresponding interpolation formulas. Finally, we give some numerical examples.

Interpolation operators on a tetrahedron with three curved edges

We construct Lagrange, Hermite and Birkhoff-type operators, which interpolate a given function and some of its derivatives on the border of a tetrahedron with three straight edges and three curved edges; we consider as well some of their product and boolean sum operators. We study the interpolation properties and the order of accuracy of the constructed operators. Finally, we give some applications and numerical examples.