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14 pages, 258 KiB

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 597

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 KiB

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

Viewed by 302

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 KiB

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 3 | Viewed by 398

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 KiB

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 9 | Viewed by 1024

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 KiB

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 1137

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 KiB

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 1 | Viewed by 1146

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 KiB

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 1013

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 KiB

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 10 | Viewed by 1944

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 KiB

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 10 | Viewed by 1763

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 KiB

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 3 | Viewed by 1742

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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13 pages, 342 KiB

Open AccessArticle

Kantorovich Type Generalization of Bernstein Type Rational Functions Based on (p,q)-Integers

by Hayatem Hamal and Pembe Sabancigil

Symmetry 2022, 14(5), 1054; https://doi.org/10.3390/sym14051054 - 20 May 2022

Cited by 5 | Viewed by 1790

Abstract

In this paper, we define a new Kantorovich-type

(p, q)

-generalization of the Balázs–Szabados operators. We derive a recurrence formula, and with the help of this formula, we give explicit formulas for the first and second-order moments, which follow a [...] Read more.

In this paper, we define a new Kantorovich-type

(p, q)

-generalization of the Balázs–Szabados operators. We derive a recurrence formula, and with the help of this formula, we give explicit formulas for the first and second-order moments, which follow a symmetric pattern. We estimate the second and fourth-order central moments. We examine the local approximation properties in terms of modulus of continuity, we give a Voronovskaja type theorem, and we give the weighted approximation properties of the operators. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory II)

20 pages, 9111 KiB

Open AccessArticle

Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters λ and α

by Qing-Bo Cai, Khursheed J. Ansari, Merve Temizer Ersoy and Faruk Özger

Mathematics 2022, 10(7), 1149; https://doi.org/10.3390/math10071149 - 2 Apr 2022

Cited by 26 | Viewed by 2324

Abstract

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters

α

and

λ

and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem [...] Read more.

This paper is devoted to studying the statistical approximation properties of a sequence of univariate and bivariate blending-type Bernstein operators that includes shape parameters

α

and

λ

and a positive integer. An estimate of the corresponding rates was obtained, and a Voronovskaja-type theorem is given by a weighted A-statistical convergence. A Korovkin-type theorem is provided for the univariate and bivariate cases of the blending-type operators. Moreover, the convergence behavior of the univariate and bivariate new blending basis and new blending operators are exhaustively demonstrated by computer graphics. The studied univariate and bivariate blending-type operators reduce to the well-known Bernstein operators in the literature for the special cases of shape parameters

α

and

λ,

and they propose better approximation results. Full article

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

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18 pages, 309 KiB

Open AccessArticle

Some New Results on Bicomplex Bernstein Polynomials

by Carlo Cattani, Çíğdem Atakut, Özge Özalp Güller and Seda Karateke

Mathematics 2021, 9(21), 2748; https://doi.org/10.3390/math9212748 - 29 Oct 2021

Cited by 2 | Viewed by 1960

Abstract

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact

C_{2}

-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate [...] Read more.

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact

C_{2}

-disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)

12 pages, 267 KiB

Open AccessArticle

Convergence of Certain Baskakov Operators of Integral Type

by Marius Mihai Birou, Carmen Violeta Muraru and Voichiţa Adriana Radu

Symmetry 2021, 13(9), 1747; https://doi.org/10.3390/sym13091747 - 19 Sep 2021

Cited by 2 | Viewed by 1817

Abstract

In the present paper, we propose a Baskakov operator of integral type using a function

φ

on

[0, \infty)

with the properties:

φ (0) = 0,

φ^{'} > 0

on

[0, \infty)

and [...] Read more.

In the present paper, we propose a Baskakov operator of integral type using a function

φ

on

[0, \infty)

with the properties:

φ (0) = 0,

φ^{'} > 0

on

[0, \infty)

and

{lim}_{x \to \infty} φ (x) = \infty

. The proposed operators reproduce the function

φ

and constant functions. For the constructed operator, some approximation properties are studied. Voronovskaja asymptotic type formulas for the proposed operator and its derivative are also considered. In the last section, the interest is focused on weighted approximation properties, and a weighted convergence theorem of Korovkin’s type on unbounded intervals is obtained. The results can be extended on the interval

(- \infty, 0]

(the symmetric of the interval

[0, \infty)

from the origin). Full article

(This article belongs to the Special Issue New Directions in Theory of Approximation and Related Problems)

12 pages, 258 KiB

Open AccessArticle

A Definition of Two-Dimensional Schoenberg Type Operators

by Camelia Liliana Moldovan and Radu Păltănea

Symmetry 2020, 12(8), 1364; https://doi.org/10.3390/sym12081364 - 17 Aug 2020

Cited by 1 | Viewed by 1982

Abstract

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms [...] Read more.

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity. Full article

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