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A Bernstein‐Like Trigonometric Basis: Properties, Curve Design, and Operator Construction

Author

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  • Jamshid Saeidian
  • Bahareh Nouri
  • Aram Azizi

Abstract

We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier‐like curves, termed “trigo‐curves”, which retain the fundamental properties of classical Bézier curves while offering enhanced shape control through parameter adjustment, independent of control point positions. We establish the monotonicity‐preserving nature of these curves. Additionally, we develop a new class of linear positive operators based on these trigonometric basis functions. The operators, incorporating an auxiliary parameter, are thoroughly analyzed for their fundamental properties. We establish their convergence rate, derive a modified Voronovskaja theorem, and obtain error bounds in terms of the modulus of continuity. Furthermore, the monotonicity‐preserving properties of these operators are also investigated.

Suggested Citation

  • Jamshid Saeidian & Bahareh Nouri & Aram Azizi, 2025. "A Bernstein‐Like Trigonometric Basis: Properties, Curve Design, and Operator Construction," Journal of Applied Mathematics, John Wiley & Sons, vol. 2025(1).
    • Handle: RePEc:wly:jnljam:v:2025:y:2025:i:1:n:5676548
    DOI: 10.1155/jama/5676548
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    References listed on IDEAS

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    1. Qing-Bo Cai & Khursheed J. Ansari & Merve Temizer Ersoy & Faruk Özger, 2022. "Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters λ and α," Mathematics, MDPI, vol. 10(7), pages 1-20, April.
    2. Sidra Maqsood & Muhammad Abbas & Gang Hu & Ahmad Lutfi Amri Ramli & Kenjiro T. Miura, 2020. "A Novel Generalization of Trigonometric Bézier Curve and Surface with Shape Parameters and Its Applications," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-25, May.
    3. Moavia Ameer & Muhammad Abbas & Kenjiro T. Miura & Abdul Majeed & Tahir Nazir, 2022. "Curve and Surface Geometric Modeling via Generalized Bézier-like Model," Mathematics, MDPI, vol. 10(7), pages 1-14, March.
    4. Moavia Ameer & Muhammad Abbas & Thabet Abdeljawad & Tahir Nazir, 2022. "A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters," Mathematics, MDPI, vol. 10(3), pages 1-19, January.
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