Kantorovich Variant Research Articles

On Kantorovich variant of Brass-Stancu operators

Abstract In this study, we deal with Kantorovich-type generalization of the Brass-Stancu operators. For the sequence of these operators, we study L p {L}^{p} -convergence and give some upper estimates for the L p {L}^{p} -norm of the approximation error via first-order averaged modulus of smoothness and the first-order K K -functional. Moreover, we show that the Kantorovich generalization of each Brass-Stancu operator satisfies variation detracting property and is bounded with respect to the norm of the space of functions of bounded variation on [ 0 , 1 ] \left[0,1] . Finally, we present graphical and numerical examples to compare the convergence of these operators to given functions under different parameters.

Neural network Kantorovich operators activated by smooth ramp functions

In the present article, we introduce a Kantorovich variant of the neural network interpolation operators activated by smooth ramp functions proposed by Qian and Yu (2022). We discuss the convergence of these operators in the spaces, and , and establish some direct approximation theorems. Further, we derive the converse results by means of Berens–Lorentz lemma and Peetre's K‐functional. We present a multivariate version of the aforementioned Kantorovich neural network interpolation operators and investigate the direct and converse results in the continuous and , spaces.

Convergence results in Orlicz spaces for sequences of max-product sampling Kantorovich operators

In this paper, we provide a unifying theory concerning the convergence properties of the so-called max-product sampling Kantorovich operators based upon generalized kernels in the setting of Orlicz spaces. The approximation of functions defined on both bounded intervals and on the whole real axis has been considered. Here, under suitable assumptions on the kernels, considered in order to define the operators, we are able to establish a modular convergence theorem for these sampling-type operators. As a direct consequence of the main theorem of this paper, we obtain that the involved operators can be successfully used for approximation processes in a wide variety of functional spaces, including the well-known interpolation and exponential spaces. This makes the Kantorovich variant of max-product sampling operators suitable for reconstructing not necessarily continuous functions (signals) belonging to a wide range of functional spaces. Finally, several examples of Orlicz spaces and of kernels for which the above theory can be applied, are presented.

Approximation properties and q-statistical convergence of Kantorovich variant of Stancu type Lupaş operators

We introduce Kantorovich variant of Stancu-Lupas? operators and study convergence and qstatistical convergence properties using Korovkin theorem. Rate of convergence is analyzed in terms of modulus of continuity, elements of Lipschitz class and Peetre?s K-functional. Direct theorems are proved and Voronovskaja type theorem is established. Graphical analysis of convergence and error estimations are presented with the help of MATLAB.

MODIFIED LUPAŞ–KANTOROVICH OPERATORS WITH PÓLYA DISTRIBUTION

We present a modification of the Kantorovich variant of the operators due to Lupaş and Lupaş which are connected with Pólya distribution and we provide a quantitative Voronovskaya-kind theorem involving modulus of continuity. Also, we observe that a better approximation can be achieved for our modified form of operators. We also estimate the difference of the original operator with its modified form. Lastly, we depict the convergence of the modified operators to some functions using graphical representation.

Construction of q-Baskakov Operators by Wavelets and Approximation Properties

We use wavelets to define the Kantorovich variant of q-Baskakov type operators, and for 1le p< infty, we study the L_{p}-approximation. Let xi be any positive constant and Psi _{k}(x) be any continuous derivative function such that int _{mathbb {R}}x^{s}Psi _{k}(x)mathrm {d}_{q}x=0 where 0le s le k,;kin mathbb {N}, 0<q<1. For all Psi in L_{infty }(mathbb {R}) suppose the following conditions hold: (i) a finite positive xi exits with the property sup Psi subset [0,xi ], (ii) its first k moments vanish: For 1le s le k,;kin mathbb {N}, we have int _{mathbb {R}}t^{s}Psi (t)mathrm {d}_{q}t=0 and int _{mathbb {R} }Psi (t)mathrm {d}_{q}t=1. Then in the sense of Haar basis for 0<q<1, the q-analogue of Baskakov–Kantorovich type wavelets operators are defined by begin{aligned} left( mathcal {S}_{r,s,q};gright) (x)=[r]_{q}sum_{s =0}^{infty }q^{s -1}B_{r,s,q}(x)int _{mathbb {R}}gleft( tright) Psi left( q^{s-1}[r]_{q}t-[s ]_{q}right) mathrm {d}_{q}t. end{aligned}

Multivariate fuzzy neural network interpolation operators and applications to image processing

In this paper, we introduce a novel family of multivariate fuzzy neural network interpolation operators activated by sigmoidal functions belonging to the new class of multivariate sigmoidal functions. To present an alternative way to the well-known shortcomings of the Hukuhara difference, we use a proper function defined on a set of fuzzy n-cell numbers. Moreover, we construct the Kantorovich variant of fuzzy NN interpolation operators, and also achieve approximation properties via Lp-type metric with respect to both modulus of continuity and Lp-modulus of continuity in fuzzy sense. Various special examples for the class of multivariate sigmoidal functions are presented. Also, we give some illustrative examples to demonstrate the approximation performances of all the above operators. Finally, we give a novel interpolation algorithm involving a multidimensional fuzzy inference system with applications in color image resizing and inpainting.

Some Properties of Kantorovich Variant of Szász Operators Induced by Multiple Sheffer Polynomials

Some Properties of Kantorovich Variant of Szász Operators Induced by Multiple Sheffer Polynomials

Approximation by Kantorovich-type max-min operators and its applications

In this study, we construct Kantorovich variant of max-min kind operators, which are nonlinear. By using these new operators, we obtain some uniform approximation results in N-dimension (N≥1). Then, we estimate the error with the help of Hölder continuous functions and modulus of continuity. Furthermore, we give some illustrative applications to verify our theory and also investigate some shape-preserving properties of Kantorovich-type max-min Bernstein operator. Lastly, we examine the image processing implementation of our results via Kantorovich-type max-min Shepard operator.

Approximation on a new class of Szász–Mirakjan operators and their extensions in Kantorovich and Durrmeyer variants with applicable properties

Abstract This paper deals with the approximation properties of a generalized version of Szász–Mirakjan operators which preserve a x {a^{x}} , a &gt; 1 {a&gt;1} (fixed), and x ≥ 0 {x\geq 0} . The uniform convergence of the operators is studied by using some auxiliary results. Also, error estimations are determined by considering the functions from different spaces. The convergence of the said operators is shown and analyzed by graphics. In the same direction, the proposed operators are compared with Szász–Mirakjan operators for the rate of convergence. A Voronovskaya-type theorem is considered and a comparison with Szász–Mirakjan operators is shown in the sense of convexity. To describe the quantitative means of an asymptotic formula, we quantitatively approach the Voronovskaya-type theorem; moreover, a Grüss–Voronovskaya-type theorem is proved. For further investigations regarding the approximation for functions from various spaces, two significant extensions are added keeping in mind some developments in the L p {L_{p}} -space. One is the Kantorovich variant and the other one is the Durrmeyer modification of the defined operators. Here, the rate of convergence is described by means of the function with a derivative of bounded variation for the Durrmeyer modified operators for which some properties are discussed. Also, A-statistical properties of the Durrmeyer modified operators are established. This paper ends by presenting some significant statements and the conclusion.

Kantorovich variant of Stancu operators

Stancu type operators play a crucial role in convergence estimates. The present article concerns the convergence estimates for certain Stancu type Kantorovich operators. We first establish some direct formulas giving the local approximation theorem, Voronovskaja type asymptotic formula, bound for the second central moment with some curtailment, and the global approximation theorem by means of modulus of continuity and the Ditzian-Totik Modulus of smoothness. We also study the difference estimates between Stancu-Bernstein operators and its Kantorovich variant. Further, we show the convergence of these operators by graphics to certain functions.

Approximation properties of the new type generalized Bernstein-Kantorovich operators

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.&lt;/p&gt;&lt;/abstract&gt;

On Kantorovich variant of Baskakov type operators preserving some functions

This paper deals with a generalization of Kantorovich variant of Baskakov type operators preserving constant function and e-2y. We discuss uniform convergence properties and weighted approximation for this generalized Baskakov-Kantorovich type operators.

Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

&lt;p style='text-indent:20px;'&gt;The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \ddot{o} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.&lt;/p&gt;

Approximation by generalized Szász-Mirakjan-Kantorovich type operators

We construct Kantorovich variant of generalized Sz?sz-Mirakjan operators whose construction depends on a continuously differentiable, increasing and unbounded function ?. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.

Operators based on Pólya distribution and finite differences

The several generalizations of the well‐known Bernstein polynomials are available in the literature. The general form of Bernstein polynomial based on Pólya distribution and its Kantorovich variant available in the literature have no links. In order to introduce a modification of a known operator, it is important to have some connection with the original one. Here, we provide a link between the such operators and its Kantorovich type integral variant using the method of backward difference operator and find another generalization, different than the one proposed before by Razi. We also provide some direct results for the Kantorovich operators. Finally, we also introduce the higher order Pólya–Kantorovich operators.

Approximation by a new sequence of operators involving Apostol-Genocchi polynomials

Abstract The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

On Approximation Properties of Generalized Lupas Type Operators Based on Polya Distribution with Pochhammer k-Symbol

The purpose of this paper is to introduce a Kantorovich variant of Lupa\c{s}-Stancu operators based on Polya distribution with Pochhammer $k$-symbol. We obtain rates of convergence for these operators by means of the classical modulus of continuity. Also, we give a Voronovskaja type theorem for the pointwise approximation. Furthermore, we construct a bivariate generalization of these operators and we discuss some convergence properties of them. Finally, we present some figures to compare approximation properties of our new operators with those of other operators which are mentioned in this paper. We observe that the approximation of our operators to the function $f$ is better than that of some other operators in a certain range of values of $k$.

Higher order Lupaş-Kantorovich operators and finite differences

It is well-known that Lupas operators are not exponential type operators, although they preserve linear functions. The differential operator is not applicable to estimate nice representation between Lupas operators and its Kantorovich variant. In the present paper we use finite difference method viz. backward differences to estimate the recurrence relation for moments of the Lupas operators. We introduce higher order Lupas-Kantorovich operators and estimate some direct results.

Kantorovich Variant of Jain-Pethe Operators

In the present paper we establish a link between the Jain-Pethe operators and its Kantorovich type integral modification, by applying the methods of finite differences. We also consider a modification of the Kantorovich variant, preserving affine functions and establish some direct results in weighted spaces.