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19 pages, 371 KB

Open AccessArticle

Adjoint Bernoulli’s Kantorovich–Schurer-Type Operators: Univariate Approximations in Functional Spaces

by Harun Çiçek, Nadeem Rao, Mohammad Ayman-Mursaleen and Sunny Kumar

Mathematics 2026, 14(2), 276; https://doi.org/10.3390/math14020276 - 12 Jan 2026

Viewed by 212

In this work, we first establish a new connection between adjoint Bernoulli’s polynomials and gamma function as a new sequence of linear positive operators denoted by

S_{r, ς, λ} (.; .)

. Further, convergence results for these [...] Read more.

In this work, we first establish a new connection between adjoint Bernoulli’s polynomials and gamma function as a new sequence of linear positive operators denoted by

S_{r, ς, λ} (.; .)

. Further, convergence results for these sequences of operators, i.e.,

S_{r, ς, λ} (.; .)

are derived in various functional spaces with the aid of the Korovkin theorem, the Voronovskaja-type theorem, the first order of the modulus of continuity, the second order of the modulus of continuity, Peetre’s K-functional, the Lipschitz condition, etc. In the last section, we focus our research on the bivariate extension of these sequences of operators; their uniform rate of approximation and order of approximation are investigated in different functional spaces. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing for Applied Mathematics)

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31 pages, 777 KB

Open AccessArticle

On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures

by Yeşim Çiçek, Rubayyi T. Alqahtani, Nezihe Turhan and Faruk Özger

Symmetry 2025, 17(12), 2139; https://doi.org/10.3390/sym17122139 - 12 Dec 2025

Viewed by 320

Abstract

In this paper, we introduce and investigate a novel family of parameter-dependent operators incorporating multiple shape parameters

{\{λ_{k}\}}_{k = 1}^{n}

, whose underlying basis functions include these parameters and exhibit a symmetry property. This parameter-dependent formulation provides a unified and [...] Read more.

In this paper, we introduce and investigate a novel family of parameter-dependent operators incorporating multiple shape parameters

{\{λ_{k}\}}_{k = 1}^{n}

, whose underlying basis functions include these parameters and exhibit a symmetry property. This parameter-dependent formulation provides a unified and flexible framework for constructing positive linear operators with enhanced approximation and shape-preserving capabilities. We establish fundamental properties of the proposed operators, including nonnegativity, linearity, end-point interpolation, monotonicity preservation and partition of unity; derive their central moments; and determine direct approximation theorems and Voronovskaja-type results. Finally, numerical experiments and graphical illustrations demonstrate the improved performance and adaptability of the proposed scheme compared with existing Bernstein-type variants. The presented framework unifies several classical and generalized operator families while providing additional shape control for practical applications in computer-aided geometric design and function approximation. Full article

(This article belongs to the Section Mathematics)

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14 pages, 335 KB

Open AccessArticle

Degenerate Appell Polynomials Connecting Beta Function as a Family of Operators and Their Approximations

by Mohammad Farid, Harun Çiçek, Mohammad Ayman-Mursaleen and Nadeem Rao

Symmetry 2025, 17(12), 2050; https://doi.org/10.3390/sym17122050 - 1 Dec 2025

Cited by 1 | Viewed by 348

Abstract

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as [...] Read more.

This paper introduces a novel family of positive linear operators constructed by blending degenerate Appell polynomials with a classical Beta kernel in the Durrmeyer setting. The operators are defined as

H_{n} (g; u) = \sum_{ƿ = 0}^{\infty} h_{ƿ} (n + u; ƛ) \int_{0}^{1} K_{n} (ƿ, ʈ) g (ʈ) d ʈ

, where

h_{ƿ} (n u; ƛ)

is derived from degenerate Appell polynomials (

n u

denotes the product of n and u) and

K_{n} (ƿ, ʈ)

is a Beta-type kernel. We establish the linearity and positivity of these operators and derive crucial moment estimates. Approximation properties are examined via Korovkin-type theorems, and the asymptotic behavior is investigated through a Voronovskaja-type theorem. The results extend and unify earlier work on Appell-based approximation operators and offer new tools for approximating functions in weighted spaces. Numerical examples and error estimates are provided to illustrate the efficacy of the proposed operators. In addition, the inherent symmetry in the structure of the proposed operators-arising from the symmetric nature of the Beta kernel and the generating functions of degenerate Appell polynomials is discussed. Such symmetry plays a key role in ensuring balanced approximation and convergence characteristics. Full article

(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)

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22 pages, 539 KB

Open AccessArticle

Kantorovich Extension of Parametric Generalized q-Schurer Operators and Their Approximation Properties

by Md. Nasiruzzaman and Abdullah Alotaibi

Mathematics 2025, 13(23), 3770; https://doi.org/10.3390/math13233770 - 24 Nov 2025

Viewed by 294

Abstract

This paper aims to extend, within the context of quantum calculus, the

α

-Bernstein–Schurer operators (

α \in [0, 1]

) to Kantorovich form. Using the Ditzian–Totik modulus of continuity and the Lipschitz-kind maximal function for our recently extended operators, [...] Read more.

This paper aims to extend, within the context of quantum calculus, the

α

-Bernstein–Schurer operators (

α \in [0, 1]

) to Kantorovich form. Using the Ditzian–Totik modulus of continuity and the Lipschitz-kind maximal function for our recently extended operators, we first examine the Korovkin-type theorem before studying the global approximation and rate of convergence, respectively. Furthermore, Taylor’s formula is used to present Voronovskaja-type theorems. Lastly, the aforementioned operators are validated through the numerical results with graphical representation. Full article

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22 pages, 460 KB

Open AccessArticle

Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results

by Md. Nasiruzzaman, Mohammad Farid and Nadeem Rao

Mathematics 2025, 13(22), 3586; https://doi.org/10.3390/math13223586 - 8 Nov 2025

Viewed by 418

Abstract

In this paper, we primarily use Stancu variants of Kantorovich-type operators to investigate the convergence and other associated properties of new Szász–Mirakjan operators. We compute the moments and central moments of the new Szász–Mirakjan operators by q-integers and propose their modified Kantorovich [...] Read more.

In this paper, we primarily use Stancu variants of Kantorovich-type operators to investigate the convergence and other associated properties of new Szász–Mirakjan operators. We compute the moments and central moments of the new Szász–Mirakjan operators by q-integers and propose their modified Kantorovich form. More specifically, we examine the convergence characteristics in the space of continuous functions. With the use of the modulus of continuity and the integral modulus of continuity, we determine the degree of convergence. Additionally, we obtain the Voronovskaja type theorems. To validate convergence, we conclude with a numerical example and graphical illustration of the operator sequences. Full article

(This article belongs to the Special Issue Advances in Functional Analysis and Approximation Theory)

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14 pages, 258 KB

Open AccessArticle

Approximation by Bicomplex Favard–Szász–Mirakjan Operators

by George A. Anastassiou, Özge Özalp Güller, Mohd Raiz and Seda Karateke

Mathematics 2025, 13(11), 1830; https://doi.org/10.3390/math13111830 - 30 May 2025

Viewed by 946

Abstract

The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact

C_{2}

disk. We provide quantitative estimates of the convergence. Moreover, the Voronovskaja-type results for analytic functions and the simultaneous approximation by bicomplex Favard–Szász–Mirakjan [...] Read more.

The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact

C_{2}

disk. We provide quantitative estimates of the convergence. Moreover, the Voronovskaja-type results for analytic functions and the simultaneous approximation by bicomplex Favard–Szász–Mirakjan operators are investigated. Full article

19 pages, 392 KB

Open AccessArticle

Szász–Beta Operators Linking Frobenius–Euler–Simsek-Type Polynomials

by Nadeem Rao, Mohammad Farid and Shivani Bansal

Axioms 2025, 14(6), 418; https://doi.org/10.3390/axioms14060418 - 29 May 2025

Cited by 1 | Viewed by 656

Abstract

This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e.,

L^{p} [0, \infty)

, [...] Read more.

This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e.,

L^{p} [0, \infty)

,

1 \leq p < \infty

. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre’s K-functional, Lipschitz type space, and the

r^{t h}

-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically. Full article

(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)

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15 pages, 660 KB

Open AccessArticle

Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

by Nadeem Rao, Mohammad Farid and Mohd Raiz

Axioms 2025, 14(4), 252; https://doi.org/10.3390/axioms14040252 - 27 Mar 2025

Cited by 11 | Viewed by 1009

Abstract

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the [...] Read more.

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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15 pages, 731 KB

Open AccessArticle

A Study of Szász–Durremeyer-Type Operators Involving Adjoint Bernoulli Polynomials

by Nadeem Rao, Mohammad Farid and Rehan Ali

Mathematics 2024, 12(23), 3645; https://doi.org/10.3390/math12233645 - 21 Nov 2024

Cited by 14 | Viewed by 1354

Abstract

This research work introduces a connection of adjoint Bernoulli’s polynomials and a gamma function as a sequence of linear positive operators. Further, the convergence properties of these sequences of operators are investigated in various functional spaces with the aid of the Korovkin theorem, [...] Read more.

This research work introduces a connection of adjoint Bernoulli’s polynomials and a gamma function as a sequence of linear positive operators. Further, the convergence properties of these sequences of operators are investigated in various functional spaces with the aid of the Korovkin theorem, Voronovskaja-type theorem, first order of modulus of continuity, second order of modulus of continuity, Peetre’s K-functional, Lipschitz condition, etc. In the last section, we extend our research to a bivariate case of these sequences of operators, and their uniform rate of approximation and order of approximation are investigated in different functional spaces. Moreover, we construct a numerical example to demonstrate the applicability of our results. Full article

(This article belongs to the Section E: Applied Mathematics)

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16 pages, 457 KB

Open AccessArticle

Approximation Properties of Chlodovsky-Type Two-Dimensional Bernstein Operators Based on (p, q)-Integers

by Ümit Karabıyık, Adem Ayık and Ali Karaisa

Symmetry 2024, 16(11), 1503; https://doi.org/10.3390/sym16111503 - 9 Nov 2024

Viewed by 1347

Abstract

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the

(p, q)

-integer. By leveraging the inherent symmetry properties of

(p, q)

-integers, we examine the approximation properties of our new operator with [...] Read more.

In the present study, we introduce the two-dimensional Chlodovsky-type Bernstein operators based on the

(p, q)

-integer. By leveraging the inherent symmetry properties of

(p, q)

-integers, we examine the approximation properties of our new operator with the help of a Korovkin-type theorem. Further, we present the local approximation properties and establish the rates of convergence utilizing the modulus of continuity and the Lipschitz-type maximal function. Additionally, a Voronovskaja-type theorem is provided for these operators. We also investigate the weighted approximation properties and estimate the rate of convergence in the same space. Finally, illustrative graphics generated with Maple demonstrate the convergence rate of these operators to certain functions. The optimization of approximation speeds by these symmetric operators during system control provides significant improvements in stability and performance. Consequently, the control and modeling of dynamic systems become more efficient and effective through these symmetry-oriented innovative methods. These advancements in the fields of modeling fractional differential equations and control theory offer substantial benefits to both modeling and optimization processes, expanding the range of applications within these areas. Full article

(This article belongs to the Section Mathematics)

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14 pages, 459 KB

Open AccessArticle

On the Properties of the Modified λ-Bernstein-Stancu Operators

by Zhi-Peng Lin, Gülten Torun, Esma Kangal, Ülkü Dinlemez Kantar and Qing-Bo Cai

Symmetry 2024, 16(10), 1276; https://doi.org/10.3390/sym16101276 - 27 Sep 2024

Cited by 2 | Viewed by 1419

Abstract

In this study, a new kind of modified

λ

-Bernstein-Stancu operators is constructed. Compared with the original

λ

-Bézier basis function, the newly operator basis function is more concise in form and has certain symmetry beauty. The moments and central moments are computed. [...] Read more.

In this study, a new kind of modified

λ

-Bernstein-Stancu operators is constructed. Compared with the original

λ

-Bézier basis function, the newly operator basis function is more concise in form and has certain symmetry beauty. The moments and central moments are computed. A Korovkin-type approximation theorem is presented, and the degree of convergence is estimated with respect to the modulus of continuity, Peetre’s K-functional, and functions of the Lipschitz-type class. Moreover, the Voronovskaja type approximation theorem is examined. Finally, some numerical examples and graphics to show convergence are presented. Full article

(This article belongs to the Section Mathematics)

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15 pages, 364 KB

Open AccessArticle

Approximation Properties of Parametric Kantorovich-Type Operators on Half-Bounded Intervals

by Hui Dong and Qiulan Qi

Mathematics 2023, 11(24), 4997; https://doi.org/10.3390/math11244997 - 18 Dec 2023

Cited by 1 | Viewed by 1205

Abstract

The main purpose of this paper is to introduce a new family of parametric Kantorovichtype operators on the half-bounded interval. The convergence properties of these new operators are investigated. The Voronovskaja-type weak inverse theorem and the rate of uniform convergence are obtained. Furthermore, [...] Read more.

The main purpose of this paper is to introduce a new family of parametric Kantorovichtype operators on the half-bounded interval. The convergence properties of these new operators are investigated. The Voronovskaja-type weak inverse theorem and the rate of uniform convergence are obtained. Furthermore, we obtain some shape preserving properties of these operators, including monotonicity, convexity, starshapeness, and semi-additivity preserving properties. Finally, some numerical illustrative examples show that these new operators have a better approximation performance than the classical ones. Full article

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19 pages, 360 KB

Open AccessArticle

A Generalization of Szász–Mirakyan Operators Based on α Non-Negative Parameter

by Khursheed J. Ansari and Fuat Usta

Symmetry 2022, 14(8), 1596; https://doi.org/10.3390/sym14081596 - 3 Aug 2022

Cited by 17 | Viewed by 2247

Abstract

The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say

α

. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows [...] Read more.

The main purpose of this paper is to define a new family of Szász–Mirakyan operators that depends on a non-negative parameter, say

α

. This new family of Szász–Mirakyan operators is crucial in that it includes both the existing Szász–Mirakyan operator and allows the construction of new operators for different values of

α

. Then, the convergence properties of the new operators with the aid of the Popoviciu–Bohman–Korovkin theorem-type property are presented. The Voronovskaja-type theorem and rate of convergence are provided in a detailed proof. Furthermore, with the help of the classical modulus of continuity, we deduce an upper bound for the error of the new operator. In addition to these, in order to show that the convex or monotonic functions produced convex or monotonic operators, we obtain shape-preserving properties of the new family of Szász–Mirakyan operators. The symmetry of the properties of the classical Szász–Mirakyan operator and of the properties of the new sequence is investigated. Moreover, we compare this operator with its classical correspondence to show that the new one has superior properties. Finally, some numerical illustrative examples are presented to strengthen our theoretical results. Full article

(This article belongs to the Special Issue Advances in Matrix Transformations, Operators and Symmetry)

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14 pages, 303 KB

Open AccessArticle

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

by Syed Abdul Mohiuddine, Arun Kajla and Abdullah Alotaibi

Mathematics 2022, 10(13), 2222; https://doi.org/10.3390/math10132222 - 25 Jun 2022

Cited by 12 | Viewed by 1966

Abstract

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative [...] Read more.

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constructed operators. Moreover, we investigate the approximation of functions with derivatives of bounded variation (DBV) of the aforesaid operators. Full article

(This article belongs to the Special Issue Higher Transcendental Functions and Their Multi-Disciplinary Applications)

13 pages, 376 KB

Open AccessArticle

Modified Bernstein–Durrmeyer Type Operators

by Arun Kajla and Dan Miclǎuş

Mathematics 2022, 10(11), 1876; https://doi.org/10.3390/math10111876 - 30 May 2022

Cited by 5 | Viewed by 1985

Abstract

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and [...] Read more.

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and a Grüss–Voronovskaja type result. Our subsequent study focuses on a direct approximation theorem using the Ditzian–Totik modulus of smoothness and the order of approximation for functions belonging to the Lipschitz-type space. For a complete image on the quantitative estimations, we included the convergence rate for differential functions, whose derivatives were of bounded variations. In the last section of the article, we present two graphs illustrating the operator convergence. Full article

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