Kantorovich Operators Research Articles

Estimate for the difference of operators having different basis functions

Very recently Aral et al. (Anal Math Phys, 2018. https://doi.org/10.1007/s1332 ) established an estimate for the difference of two linear positive operators, but their result is valid for operators having same basis functions. In the present article, we study the approximation of difference of operators and establish a general estimate for the difference of those operators, which have different basis functions. In the end, we give examples of quantitative estimates for the difference between the Baskakov and Szasz–Mirakyan operators and between Szasz–Mirakyan and Baskakov–Kantorovich operators in terms of weighted modulus of continuity. Also, graphical representation is presented for such examples.

Quantitative Estimates for Differences of Baskakov-Type Operators

In the present paper, we establish the quantitative estimates in terms of weighted modulus of continuity, for the differences of Baskakov operators with Baskakov–Szasz operators, Baskakov–Kantorovich operators and Genuine Baskakov–Durrmeyer type operators. Also, we find the estimates for the mutual differences of these operators. Finally, we provide an estimate for the difference of operators by direct method.

A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium

In this paper, we develop an algorithm for the segmentation of the pervious lumen of the aorta artery in computed tomography (CT) images without contrast medium, a challenging task due to the closeness gray levels of the different zones to segment. The novel approach of the proposed procedure mainly resides in enhancing the resolution of the image by the application of the algorithm deduced from the mathematical theory of sampling Kantorovich operators. After the application of suitable digital image processing techniques, the pervious zone of the artery can be distinguished from the occluded one. Numerical tests have been performed using 233 CT images, and suitable numerical errors have been computed and introduced ex novo to evaluate the performance of the proposed method. The above procedure is completely automatic in all its parts after the initial region of interest (ROI) selection. The main advantages of this approach relies in the potential possibility of performing diagnosis concerning vascular pathologies even for patients with severe kidney diseases or allergic problems, for which CT images with contrast medium cannot be achieved.

Kantorovich‐type generalization of parametric Baskakov operators

In this manuscript, we define a Kantorovich generalization of the nonnegative parametric Baskakov operators. After that, the weighted uniform convergence of the generalized operators is proved. Also, we present Voronovskaja‐type asymptotic approximation theorem then establish weighted approximation properties for parametric Kantorovich operator. Numerical results show that depending on the value of the parameter, we obtain a better approximation.

Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. 41 (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

Inverse results of approximation and the saturation order for the sampling Kantorovich series

In the present paper, we study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. In order to achieve the above result, we first need to establish a relation between the sampling Kantorovich operators and the classical generalized sampling series of P.L. Butzer. Further, for the latter operators, we also need to prove a Voronovskaja-type formula. Moreover, inverse theorems of approximation are also obtained, in order to give a characterization for the rate of convergence of the above operators. Finally, several examples of kernel functions for which the above theory can be applied have been presented and discussed in details.

A Quantitative Estimate for the Sampling Kantorovich Series in Terms of the Modulus of Continuity in Orlicz Spaces

In the present paper we establish a quantitative estimate for the sampling Kantorovich operators with respect to the modulus of continuity in Orlicz spaces defined in terms of the modular functional. At the end of the paper, concrete examples are discussed, both for what concerns the kernels of the above operators, as well as for some concrete instances of Orlicz spaces.

A Characterization of the Absolute Continuity in Terms of Convergence in Variation for the Sampling Kantorovich Operators

We study the convergence in variation for the sampling Kantorovich operators in both the cases of averaged-type kernels and classical band-limited kernels. In the first case, a characterization of the space of the absolutely continuous functions in terms of the convergence in variation is obtained.

A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators.

Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators

In this study, we consider statistical approximation properties of univariate and bivariate ?-Kantorovich operators. We estimate rate of weighted A-statistical convergence and prove a Voronovskajatype approximation theorem by a family of linear operators using the notion of weighted A-statistical convergence. We give some estimates for differences of ?-Bernstein and ?-Durrmeyer, and ?-Bernstein and ?-Kantorovich operators. We establish a Voronovskaja-type approximation theorem by weighted A-statistical convergence for the bivariate case.

Uniform boundedness of Kantorovich operators in variable exponent Lebesgue spaces

In this paper, the Kantorovich operators Kn, n ? N are shown to be uniformly bounded in variable exponent Lebesgue spaces on the closed interval [0; 1]. Also an upper estimate is obtained for the difference Kn(f)-f for functions f of regularity of order 1 and 2 measured in variable exponent Lebesgue spaces, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

Improved approximation and error estimations by King type (p, q)-Szász-Mirakjan Kantorovich operators

In the present paper, two different modifications are proposed for (p, q)-Szász-Mirakjan-Kantorovich operators which preserve some test functions. Some approximation results with the help of better-known Korovkin’s theorem and weighted Korovkin’s theorem for these operators are presented. Furthermore, convergence properties in terms of modulus of continuity and class of Lipschitz functions are studied. It has been shown that for a given absolute error bound, King type modified (p, q)-Szász-Mirakjan-Kantorovich operators require lesser value of m and elapsed time within some subintervals. Further for comparisons, some graphics and error estimation tables are presented using MATLAB(R2018a).

On difference of operators with applications to Szász type operators

In the present article, we establish the approximation of differences of operators. We obtain quantitative estimates for the difference of Szasz operators with Szasz–Baskakov, Phillips operators, Szasz–Kantorovich operators and Szasz–Durrmeyer operators. Also, their mutual differences are possible, which are estimated in the present paper.

An Inverse Result of Approximation by Sampling Kantorovich Series

AbstractIn the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

Extension of Saturation Theorems for the Sampling Kantorovich Operators

In this paper, we extend the saturation results for the sampling Kantorovich operators proved in a previous paper, to more general settings. In particular, exploiting certain Voronovskaja-formulas for the well-known generalized sampling series, we are able to extend a previous result from the space of $$C^2$$ -functions to the space of $$C^1$$ -functions. Further, requiring an additional assumption, we are able to reach a saturation result even in the space of the uniformly continuous and bounded functions. In both the above cases, the assumptions required on the kernels, which define the sampling Kantorovich operators, have been weakened with respect to those assumed previously. On this respect, some examples have been discussed at the end of the paper.

Blending type approximation by GBS operators of bivariate tensor product of \u03bb-Bernstein\u2013Kantorovich type

In this paper, we introduce a family of GBS operators of bivariate tensor product of λ-Bernstein–Kantorovich type. We estimate the rate of convergence of such operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, establish the Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators, as well as give some examples and their graphs to show the effect of convergence.

Approximation Properties of Kantorovich Type Modifications of $(p, q)-$Meyer-König-Zeller Operators

In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-König-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-König-Zeller Kantorovich operators by means of Matlab programming.

Approximation by truncated max‐product operators of Kantorovich‐type based on generalized (ϕ,ψ)‐kernels

Suggested by the max‐product sampling operators based on sinc‐Fejér kernels, in this paper, we introduce truncated max‐product Kantorovich operators based on generalized type kernels depending on two functions ϕ and ψ satisfying a set of suitable conditions. Pointwise convergence, quantitative uniform convergence in terms of the moduli of continuity, and quantitative Lp‐approximation results in terms of a K‐functional are obtained. Previous results in sampling and neural network approximation are recaptured, and new results for many concrete examples are obtained.

Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem

This work is concerned with various weighted four dimensional matrix summability methods in modular function spaces associating with generalized difference operator involving (p,q)-gamma function. Following very recent results of Kadak (2017) [20], we first introduce a new type of difference operator on double function sequences and, also define modularly weighted A-statistical convergence and modularly statistically N˜A-summability by means of weighted four dimensional regular matrix A. We also present some important inclusion relations between newly proposed summability methods. Moreover, based on the concept of modularly statistically N˜A-summability, we prove a Korovkin type approximation theorem for functions of two variables in modular spaces. Finally, in order to show that our proposed method and its applications to approximation results are stronger than the existing methods, we display an illustrative example using bivariate (p,q)-Bernstein Kantorovich operators.

Modified Kantorovich operators with better approximation properties

In the present paper, we study a new kind of Bernstein-Kantorovich-type operators. Here, we discuss a uniform convergence estimate for this modified form. Also, some direct estimates, which involve the asymptotic-type results, are established. Some numerical examples which show the relevance of the results are considered.