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 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.