Kantorovich Operators Research Articles

Uniform boundedness of Kantorovich operators in Morrey spaces

In this paper, the Kantorovich operators $$K_n, n\in \mathbb {N}$$ are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference $$K_n(f)-f$$ for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

The Bézier variant of Lupas Kantorovich operators based on Polya distribution

The Bézier variant of Lupas Kantorovich operators based on Polya distribution

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 $$[0,\infty )$$ .

Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [12], in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic differential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves.The analysis is carried out in the context of the space C(K) of all continuous functions defined on an arbitrary convex compact subset K of Rd, d≥1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1≤p<+∞. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We finally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensional hypercubes and simplices. In these settings the relevant differential operators fall into the class of Fleming–Viot operators.

Detection of thermal bridges from thermographic images by means of image processing approximation algorithms

In this paper, we develop a procedure for the detection of the contours of thermal bridges from thermographic images, in order to study the energy performance of buildings. Two main steps of the above method are: the enhancement of the thermographic images by an optimized version of the mathematical algorithm for digital image processing based on the theory of sampling Kantorovich operators, and the application of a suitable thresholding based on the analysis of the histogram of the enhanced thermographic images. Finally, an improvement of the parameter defining the thermal bridge is obtained.

Uniform and Pointwise Quantitative Approximation by Kantorovich–Choquet Type Integral Operators with Respect to Monotone and Submodular Set Functions

In this paper, for the univariate Bernstein–Kantorovich, Szasz–Mirakjan–Kantorovich and Baskakov–Kantorovich operators written in terms of the Choquet integral with respect to a monotone and submodular set function, we obtain quantitative approximation estimates, uniform and pointwise in terms of the modulus of continuity. In addition, we show that for large classes of functions, the Kantorovich–Choquet type operators approximate better than their classical correspondents. Also, we construct new Szasz–Mirakjan–Kantorovich–Choquet and Baskakov–Kantorovich–Choquet operators, which approximate uniformly f in each compact subinterval of \([0, +\infty )\) with the order \(\omega _{1}(f; \sqrt{\lambda _{n}})\), where \(\lambda _{n}\searrow 0\) arbitrary fast.

Modified Bernstein–Kantorovich operators for functions of one and two variables

The purpose of this note is to obtain some direct results for the modified Bernstein–Kantorovich operators for functions of one variable proposed by Ozarslan and Duman (Numer Funct Anal Optim 37:92–105, 2016). We also introduce the bivariate extension of these operators and compute the Voronovskaja type asymptotic formula, the degree of approximation for the Lipschitz class of elements and order of approximation by means of the Peetre’s K-functional. We illustrate the convergence of the operators to a certain function with the help of Mathematica software.

Statistical relative $$\mathcal {A}$$ A -summation process for double sequences on modular spaces

In the present paper, we extend the Korovkin type approximation theorem via statistical relative $$\mathcal {A}$$ -summation process onto the double sequences of positive linear operators in a modular space. Then we discuss the reduced results which are obtained by special choice of the scale function and the matrix sequences. We apply our new result to bivariate Bernstein–Kantorovich operators in Orlicz spaces and hence we show that it is stronger than the results obtained previously.

A note on Szász–Mirakyan–Kantorovich type operators preserving $$e^{-x}$$ e - x

In the present article, we study modified form of Szasz–Mirakyan–Kantorovich operators, which reproduce constant and \(e^{-x}\) functions. We discuss a uniform convergence result along with a quantitative estimate for the modified operators.

A generalization of Kantorovich operators for convex compact subsets

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the 1-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

Q- Lupas Kantorovich operators based on Polya distribution

The purpose of the present paper is to introduce a Kantorovich modification of the q-analogue of the Stancu operators defined by Nowak (J Math Anal Appl 350:50–55, 2009). We study a local and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Further A-statistical convergence properties of these operators are investigated. Next, a bivariate generalization of these operators is introduced and its rate of convergence is discussed with the aid of the partial and complete modulus of continuity and the Peetre‘s K-functional.

Statistical approximation to Chlodowsky type q-Bernstein-Schurer-Stancu-Kantorovich operators

In this paper we introduce two kinds of Chlodowsky-type q-Bernstein-Schurer-Stancu- Kantorovich operators on the onbounded domain. The Korovkin type statistical approximation property of these operators are investigated. We investigated the rate of convergence for this approximation by means of the first and the second modulus of continuity. The rate of convergence is investigated by using Lipschitz classes of functions and the modulus of continuity of the derivative of the function. Then, we obtain point-wise estimate the Lipchitz type maximal function.

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 &amp; Sons, Ltd.

Generalized Kantorovich Operators on Bauer Simplices and Their Limit Semigroups

ABSTRACTIn this paper, we prove an asymptotic formula for generalized Kantorovich operators associated with the canonical Markov projection on a given Bauer simplex K. That formula involves an operator acting on the subalgebra of all products of affine functions on K. Moreover, we prove that such an operator is closable and its closure is the generator of a Markov semigroup which, in turn, may be represented in terms of iterates of the above-mentioned generalized Kantorovich operators.

Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein–Stancu–Kantorovich operators

In this paper, we introduce the bivariate generalization of the Chlodowsky-type q-Bernstein–Stancu–Kantorovich operators on an unbounded domain and studied the rate of convergence in terms of the Lipschitz class function and complete modulus of continuity. Further, we establish the weighted approximation properties for these operators. The aim of this paper is to obtain the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and the Peetre’s K- functional. Then, we give generalization of the operators and investigate their approximations. Furthermore, we show the convergence of the bivariate Chlodowsky-type operators to certain functions by illustrative graphics using Python programming language. Finally, we construct the GBS operators of bivariate Chlodowsky-type q-Bernstein–Stancu–Kantorovich and estimate the rate of convergence for these operators with the help of mixed modulus of smoothness.

Approximation of discontinuous signals by sampling Kantorovich series

In this paper, the behavior of the sampling Kantorovich operators has been studied, when discontinuous functions (signals) are considered in the above sampling series. Moreover, the rate of approximation for the family of the above operators is estimated, when uniformly continuous and bounded signals are considered. Finally, several examples of (duration-limited) kernels which satisfy the assumptions of the present theory have been provided, and also the problem of the linear prediction by sampling values from the past is analyzed.

Approximation by generalized positive linear Kantorovich operators

In the present article, we introduce generalized positive linear-Kantorovich operators depending on P?lya-Eggenberger distribution (PED) as well as inverse P?lya-Eggenberger distribution (IPED) and for these operators, we study some approximation properties like local approximation theorem, weighted approximation and estimation of rate of convergence for absolutely continuous functions having derivatives of bounded variation.

Better approximation results by Bernstein–Kantorovich operators

In this paper, we give a King-type modification of the Bernstein–Kantorovich operators and study the approximation properties of these operators. We prove that the error estimation of these operators is better than the classical Bernstein–Kantorovich operators. We also give some estimations for the rate of convergence of these operators by using the modulus of continuity. Furthermore, we obtain a Voronovskaya-type asymptotic formula for these operators.

Generalized Baskakov Kantorovich operators

In this paper, we construct generalized Baskakov Kantorovich operators. We establish some direct results and then study weighted approximation, simultaneous approximation and statistical convergence properties for these operators. Finally, we obtain the rate of convergence for functions having a derivative coinciding almost everywhere with a function of bounded variation for these operators.

New integral type operators

In this paper we construct new integral type operators including heritable properties of Baskakov Durrmeyer and Baskakov Kantorovich operators. Results concerning convergence of these operators in weighted space and the hypergeometric form of the operators are shown. Voronovskaya type estimate of the pointwise convergence along with its quantitative version based on the weighted modulus of smoothness are given. Moreover, we give a direct approximation theorem for the operators in suitable weighted Lp space on [0,?).