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

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...

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Bibliographic Details
Published inJournal of applied mathematics Vol. 2025; no. 1
Main Authors Saeidian, Jamshid, Nouri, Bahareh, Azizi, Aram
Format Journal Article
LanguageEnglish
Published New York John Wiley & Sons, Inc 01.01.2025
Wiley
Subjects
Approximation
Computer graphics
Construction
Design
Flexibility
Hypotheses
Polynomials
Symmetry
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ISSN1110-757X
1687-0042
DOI10.1155/jama/5676548

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Summary: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.
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ISSN:1110-757X
1687-0042
DOI:10.1155/jama/5676548