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

Çiçek, Yeşim; Alqahtani, Rubayyi T.; Turhan, Nezihe; Özger, Faruk

doi:10.3390/sym17122139

Open AccessArticle

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

by

Yeşim Çiçek

¹

,

Rubayyi T. Alqahtani

²

,

Nezihe Turhan

¹

and

Faruk Özger

^3,*

¹

Department of Engineering Sciences, Izmir Katip Celebi University, Izmir 35620, Türkiye

²

Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia

³

Department of Computer Engineering, Igdır University, Igdır 76000, Türkiye

^*

Author to whom correspondence should be addressed.

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

Submission received: 17 October 2025 / Revised: 25 November 2025 / Accepted: 10 December 2025 / Published: 12 December 2025

(This article belongs to the Section Mathematics)

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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 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.

Keywords: divided difference; symmetric basis functions; shape-preserving approximation; convergence analysis; error graphics

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MDPI and ACS Style

Çiçek, Y.; Alqahtani, R.T.; Turhan, N.; Özger, F. On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures. Symmetry 2025, 17, 2139. https://doi.org/10.3390/sym17122139

AMA Style

Çiçek Y, Alqahtani RT, Turhan N, Özger F. On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures. Symmetry. 2025; 17(12):2139. https://doi.org/10.3390/sym17122139

Chicago/Turabian Style

Çiçek, Yeşim, Rubayyi T. Alqahtani, Nezihe Turhan, and Faruk Özger. 2025. "On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures" Symmetry 17, no. 12: 2139. https://doi.org/10.3390/sym17122139

APA Style

Çiçek, Y., Alqahtani, R. T., Turhan, N., & Özger, F. (2025). On a Family of Parameter-Dependent Bernstein-Type Operators with Multiple Shape Parameters: Incorporating Symmetric Basis Structures. Symmetry, 17(12), 2139. https://doi.org/10.3390/sym17122139

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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