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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

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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24 pages, 6364 KB

Open AccessArticle

Bezier Curves and Surfaces with the Generalized α-Bernstein Operator

by Davut Canlı and Süleyman Şenyurt

Symmetry 2025, 17(2), 187; https://doi.org/10.3390/sym17020187 - 25 Jan 2025

Cited by 1 | Viewed by 1348

Abstract

In the field of Computer-Aided Geometric Design (CAGD), a proper model can be achieved depending on certain characteristics of the predefined blending basis functions. The presence of these characteristics ensures the geometric properties necessary for a decent design. The objective of this study, [...] Read more.

α

-Bernstein operator in the context of its potential classification as a novel blending type basis for the construction of Bézier-like curves and surfaces. First, a recursive definition of this basis is provided, along with its unique representation in terms of that for the classical Bernstein operator. Next, following these representations, the characteristics of the basis are discussed, and one shape parameter for

α

-Bezier curves is defined. In addition, by utilizing the recursive definition of the basis, a de Casteljau-like algorithm is provided such that a subdivision schema can be applied to the construction of the new

α

-Bezier curves. The parametric continuity constraints for

C^{0}

C^{1}

, and

C^{2}

are also established to join two

α

-Bezier curves. Finally, a set of cross-sectional engineering surfaces is designed to indicate that the generalized

α

-Bernstein operator, as a basis, is efficient and easy to implement for forming shape-adjustable designs. Full article

(This article belongs to the Section Mathematics)

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12 pages, 284 KB

Open AccessArticle

Genuine q-Stancu-Bernstein–Durrmeyer Operators

by Pembe Sabancıgil

Symmetry 2023, 15(2), 437; https://doi.org/10.3390/sym15020437 - 7 Feb 2023

Cited by 3 | Viewed by 1608

Abstract

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators

Z_{n}^{q, α} (f; x)

. We calculate the moments of these operators,

Z_{n}^{q, α} (t^{j}; x)

for [...] Read more.

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators

Z_{n}^{q, α} (f; x)

. We calculate the moments of these operators,

Z_{n}^{q, α} (t^{j}; x)

for

j = 0, 1, 2,

which follows a symmetric pattern. We also calculate the second order central moment

Z_{n}^{q, α} ({(t - x)}^{2}; x) .

We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order

β

and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order. Full article

21 pages, 617 KB

Open AccessArticle

Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

by Faruk Özger, Ekrem Aljimi and Merve Temizer Ersoy

Mathematics 2022, 10(12), 2027; https://doi.org/10.3390/math10122027 - 11 Jun 2022

Cited by 39 | Viewed by 2784

Abstract

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials [...] Read more.

λ

α

and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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20 pages, 9111 KB

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 28 | Viewed by 2465

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, 301 KB

Open AccessArticle

Bivariate α,q-Bernstein–Kantorovich Operators and GBS Operators of Bivariate α,q-Bernstein–Kantorovich Type

by Qing-Bo Cai, Wen-Tao Cheng and Bayram Çekim

Mathematics 2019, 7(12), 1161; https://doi.org/10.3390/math7121161 - 2 Dec 2019

Cited by 7 | Viewed by 2375

Abstract

In this paper, we introduce a family of bivariate

α, q

-Bernstein–Kantorovich operators and a family of

G B S

(Generalized Boolean Sum) operators of bivariate

α, q

-Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central [...] Read more.

In this paper, we introduce a family of bivariate

α, q

-Bernstein–Kantorovich operators and a family of

G B S

(Generalized Boolean Sum) operators of bivariate

α, q

-Bernstein–Kantorovich type. For the former, we obtain the estimate of moments and central moments, investigate the degree of approximation for these bivariate operators in terms of the partial moduli of continuity and Peetre’s K-functional. For the latter, we estimate the rate of convergence of these

G B S

operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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