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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 602

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 306

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, 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 1035

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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13 pages, 490 KiB

Open AccessArticle

On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties

by Md. Heshamuddin, Nadeem Rao, Bishnu P. Lamichhane, Adem Kiliçman and Mohammad Ayman-Mursaleen

Mathematics 2022, 10(18), 3227; https://doi.org/10.3390/math10183227 - 6 Sep 2022

Cited by 20 | Viewed by 2183

Abstract

The goal of this research article is to introduce a sequence of

α

–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A Voronovskaja-type approximation result is also proven. Next, error [...] Read more.

The goal of this research article is to introduce a sequence of

α

–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A Voronovskaja-type approximation result is also proven. Next, error analysis and convergence of the operators for certain functions are presented numerically and graphically. Furthermore, two-dimensional

α

–Stancu–Schurer–Kantorovich operators are constructed and their rate of convergence, graphical representation of approximation and numerical error estimates are presented. Full article

(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)

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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 11 | Viewed by 1958

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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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 1747

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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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 2327

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, 317 KiB

Open AccessArticle

Inequalities for Approximation of New Defined Fuzzy Post-Quantum Bernstein Polynomials via Interval-Valued Fuzzy Numbers

by Esma Yıldız Özkan

Symmetry 2022, 14(4), 696; https://doi.org/10.3390/sym14040696 - 28 Mar 2022

Cited by 5 | Viewed by 1748

Abstract

In this study, we introduce new defined fuzzy post-quantum Bernstein polynomials and present examples illustrating their existence. We investigate their approximation properties via interval-valued fuzzy numbers. We obtain a fuzzy Korovkin-type approximation result, and we obtain inequalities estimating the rate of fuzzy convergence [...] Read more.

In this study, we introduce new defined fuzzy post-quantum Bernstein polynomials and present examples illustrating their existence. We investigate their approximation properties via interval-valued fuzzy numbers. We obtain a fuzzy Korovkin-type approximation result, and we obtain inequalities estimating the rate of fuzzy convergence for these polynomials by means of the fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Lastly, we present a Voronovskaja type asymptotic result for fuzzy post-quantum Bernstein polynomials. The methods in this paper are crucial and symmetric in terms of extending the approximation results of these polynomials from the real function space to the fuzzy function space and the applicability to the other operators. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

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 1966

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 1823

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)

13 pages, 321 KiB

Open AccessArticle

Modified Operators Interpolating at Endpoints

by Ana Maria Acu, Ioan Raşa and Rekha Srivastava

Mathematics 2021, 9(17), 2051; https://doi.org/10.3390/math9172051 - 25 Aug 2021

Cited by 1 | Viewed by 1835

Abstract

Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate [...] Read more.

Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate at endpoints although they do not preserve the affine functions. We investigate the properties of these modified operators and obtain results concerning iterates and their limits, Voronovskaja-type results and estimates of several differences. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

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11 pages, 269 KiB

Open AccessFeature PaperArticle

Voronovskaja-Type Quantitative Results for Differences of Positive Linear Operators

by Ana Maria Acu, Gülen Başcanbaz-Tunca and Ioan Raşa

Symmetry 2021, 13(8), 1392; https://doi.org/10.3390/sym13081392 - 31 Jul 2021

Cited by 3 | Viewed by 1853

Abstract

We consider positive linear operators having the same fundamental functions and different functionals in front of them. For differences involving such operators, we obtain Voronovskaja-type quantitative results. Applications illustrating the theoretical aspects are presented. Full article

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

13 pages, 365 KiB

Open AccessArticle

Durrmeyer-Type Generalization of Parametric Bernstein Operators

by Arun Kajla, Mohammad Mursaleen and Tuncer Acar

Symmetry 2020, 12(7), 1141; https://doi.org/10.3390/sym12071141 - 8 Jul 2020

Cited by 14 | Viewed by 2336

Abstract

In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic [...] Read more.

In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software. Full article

(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)

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13 pages, 266 KiB

Open AccessEditor’s ChoiceArticle

Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators

by Hari Mohan Srivastava, Gürhan İçöz and Bayram Çekim

Axioms 2019, 8(4), 111; https://doi.org/10.3390/axioms8040111 - 10 Oct 2019

Cited by 31 | Viewed by 3795

Abstract

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation [...] Read more.

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

22 pages, 338 KiB

Open AccessFeature PaperArticle

Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

by Hari M. Srivastava, Faruk Özger and S. A. Mohiuddine

Symmetry 2019, 11(3), 316; https://doi.org/10.3390/sym11030316 - 2 Mar 2019

Cited by 93 | Viewed by 4194

Abstract

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter

λ \in [- 1, 1]

and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. [...] Read more.

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter

λ \in [- 1, 1]

and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type

λ

-Bernstein operators and study their approximation behaviors. Full article

(This article belongs to the Special Issue Integral Transformations, Operational Calculus and Their Applications)

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