Let $ f $ be a real-valued continuous function defined on the interval $ [0, 1] $, and let $ B_{n, \lambda}(f; \cdot) $ be the associated $ n $-th degenerate Bernstein polynomial. We proved that, on any admissible interval, $ |B_{n, \lambda}(f; \cdot)-f(\cdot)| $ converges uniformly to $ 0 $ as $ n \to \infty $. We also established the uniform convergence of the first derivative of the degenerate Bernstein polynomial. To support the theoretical results, we tested the approximation theory on $ f(x) = \sin(\pi x) $ and its first-order derivative $ f'(x) = \pi\cos(\pi x) $. These showed that admissible pairs, and some formal pairs close to the admissible range, give better function approximation than the classical Bernstein case, and the first-derivative approximation convergence increases as $ \lambda_n \to 0 $.
Citation: Murad Khalil, Seongho Park, Si Hyeon Lee, Kyo-Shin Hwang. Uniform approximation for degenerate Bernstein polynomials[J]. AIMS Mathematics, 2026, 11(7): 19824-19854. doi: 10.3934/math.2026805
Let $ f $ be a real-valued continuous function defined on the interval $ [0, 1] $, and let $ B_{n, \lambda}(f; \cdot) $ be the associated $ n $-th degenerate Bernstein polynomial. We proved that, on any admissible interval, $ |B_{n, \lambda}(f; \cdot)-f(\cdot)| $ converges uniformly to $ 0 $ as $ n \to \infty $. We also established the uniform convergence of the first derivative of the degenerate Bernstein polynomial. To support the theoretical results, we tested the approximation theory on $ f(x) = \sin(\pi x) $ and its first-order derivative $ f'(x) = \pi\cos(\pi x) $. These showed that admissible pairs, and some formal pairs close to the admissible range, give better function approximation than the classical Bernstein case, and the first-derivative approximation convergence increases as $ \lambda_n \to 0 $.
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Murad Khalil, Seongho Park, Si Hyeon Lee, Kyo-Shin Hwang. Uniform approximation for degenerate Bernstein polynomials[J]. AIMS Mathematics, 2026, 11(7): 19824-19854. doi: 10.3934/math.2026805
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