Kantorovich Variant Research Articles

Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.

Kantorovich Variant of Ismail–May Operators

Kantorovich Variant of Ismail–May Operators

A Kantorovich variant of Lupaş–Stancu operators based on Pólya distribution with error estimation

In this paper, we introduce the Kantorovich variant of Lupas operators based on Polya distribution with shifted knots. The advantage of using shifted knots is that one can do approximation on [0,1] as well as on its subinterval. Also, it adds flexibility to operators for approximation. First, some basic results for convergence of the introduced operators are established and its rate of convergence is discussed with the aid of the modulus of continuity and Peetre K-functional. Further, a Voronovskaja type theorem for the said operators are studied. Next, a bivariate generalization of these operators are introduced. Some numerical examples with illustrative graphics have been added to justify the theoretical results and also compare the rate of convergence with the help of MATLAB. It has been shown that for suitable choices of parameters error estimate and elapsed time can be further minimized.

Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression

In a recent paper, for univariate max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several \begin{document}$ L^{p}_{\mu} $\end{document} convergence properties on bounded intervals or on the whole real axis. In this paper, firstly we obtain quantitative estimates with respect to a \begin{document}$ K $\end{document} -functional, for the multivariate Kantorovich variant of these max-product sampling operators with the integrals written in terms of Borel probability measures. Applications of these approximation results to learning theory are obtained.

A Kantorovich variant of a generalized Bernstein operators

A Kantorovich variant of a generalized Bernstein operators

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The rth order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.

Approximation degree of a Kantorovich variant of Stancu operators based on Polya–Eggenberger distribution

This paper is a continuation of the work done by Deo et al. (Appl. Math. Comput. 273, 281–289, 2016), in which the authors have established some approximation properties of the Stancu–Kantorovich operators based on Polya–Eggenberger distribution. We obtain some direct results for these operators by means of the Lipschitz class function, the modulus of continuity and the weighted space. Also, we study an approximation theorem with the aid of the unified Ditzian–Totik modulus of smoothness $$\omega _{\phi ^{\tau }}(f;t),\,\,\,0\le \tau \le 1$$ and the rate of convergence of the operators for the functions having a derivative which is locally of bounded variation on $$[0,\infty )$$ .

The Kantorovich variant of an operator defined by D. D. Stancu

In this note we introduce Kantorovich variant of the operators considered by Stancu (1998) based on two nonnegative parameters. Here, we prove an approximation theorem with the help of Bohman–Korovkin’s principle and find the estimate of the rate of convergence by means of modulus of smoothness and Lipschitz type function for these operators. In the last section of the paper, we show the convergence of the operators by illustrative graphics in Mathematica to certain functions.

The approximation of bivariate Chlodowsky-Sz\xe1sz-Kantorovich-Charlier-type operators

In this paper, we introduce a bivariate Kantorovich variant of combination of Szász and Chlodowsky operators based on Charlier polynomials. Then, we study local approximation properties for these operators. Also, we estimate the approximation order in terms of Peetre’s K-functional and partial moduli of continuity. Furthermore, we introduce the associated GBS-case (Generalized Boolean Sum) of these operators and study the degree of approximation by means of the Lipschitz class of Bögel continuous functions. Finally, we present some graphical examples to illustrate the rate of convergence of the operators under consideration.

Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution

Abstract In the paper the authors introduce the Kantorovich variant of Stancu operators based on Pólya-Eggenberger distribution. By making use of this new operator, we obtain some indispensable auxiliary results. We also deal with a Voronovskaja type asymptotic formula and some estimates of the rate of approximation involving modulus of smoothness, such as Ditzian-Totik modulus of smoothness. The rate of convergence for differential functions whose derivatives are bounded is also obtained.

Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators

Abstract In this paper we introduce Kantorovich variant of the Mittag-Leffler operators including the modified Kantorovich-Szász-Mirakjan operators. We give αβ-statistical approximation theorems for these operators in various function spaces. The results include the statistical, lacunary statistical and λ-statistical cases. Moreover, we compute the rate of convergence in different Lipschitz type spaces.

Approximation by a Kantorovich Variant of Szász Operators Based on Brenke-Type Polynomials

In this paper, we estimate the rate of convergence on function of bounded variation for a Kantorovich variant of a generalization of Szasz operators involving Brenke-type polynomials by means of some methods and techniques of probability theory.

On a Kantorovich Variant of(p,q)-Szász-Mirakjan Operators

We propose a Kantorovich variant of(p,q)-analogue of Szász-Mirakjan operators. We establish the moments of the operators with the help of a recurrence relation that we have derived and then prove the basic convergence theorem. Next, the local approximation and weighted approximation properties of these new operators in terms of modulus of continuity are studied.

Some approximation properties of Kantorovich variant of Chlodowsky operators based on q-integer

In this paper, we introduce two digerent Kantorovich type generalization of the q Chlodowsky operators. For the first operators we give some weighted approximation theorems and a Voronovskaja type theorem. Also, we present the local approximation properties and the order of convergence forunbounded functions of these operators . For second operators, we obtain aweighted statistical approximation property

Further Results for kth order Kantorovich Modification of Linking Baskakov Type Operators

In this paper we consider a non-trivial link between Baskakov type operators and their genuine Durrmeyer type modification as well as the kth order Kantorovich variant. Recursion formulas for the moments and the images of monomials are proved in order to derive asymptotic expansions. Furthermore we investigate convexity properties of the linking operators and the limiting behavior for certain function spaces.

A Kantorovich Type of Szasz Operators Including Brenke‐Type Polynomials

We give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke‐type polynomials and obtain convergence properties of these operators by using Korovkin′s theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre′s K‐functional. Furthermore, an example of Kantorovich type of the operators including Gould‐Hopper polynomials is presented and Voronovskaya‐type result is given for these operators including Gould‐Hopper polynomials.

On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators Jn,αf for functions f measurable and locally bounded on the interval [0,∞) with α⩾1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of Jn,αf(x) at those x>0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann–Butzer–Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.

Rate of Convergence of a New Type Kantorovich Variant of Bleimann-Butzer-Hahn Operators

A new type Kantorovich variant of Bleimann-Butzer-Hahn operator is introduced. Furthermore, the approximation properties of the operators are studied. An estimate on the rate of convergence of the operators for functions of bounded variation is obtained.

On Kantorovich Process of a Sequence of the Generalized Linear Positive Operators

We define the Kantorovich variant of the generalized linear positive operators introduced by Ibragimov and Gadjiev in 1970. We investigate direct approximation result for these operators on p-weighted integrable function spaces and also estimate their rate of convergence for absolutely continuous functions having a derivative coinciding a.e., with a function of bounded variation.

Approximation by a Kantorovich variant of the Bleimann, Butzer and Hahn operators

We study the approximation properties of a Kantorovich variant of the Bleimann, Butzer and Hahn operators for locally bounded functions, and estimate their rate of convergence by some techniques of probability theory and analysis methods. Mathematics subject classification (2000): 41A36, 41A25, 41A10.