@article{Khalil2026, 
author = {Murad Khalil and Seongho Park and Si Hyeon Lee and Kyo-Shin Hwang},
title = {Uniform approximation for degenerate Bernstein polynomials},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {7},
pages = {19824-19854},
keywords = {degenerate Bernstein polynomial, uniform convergence, modulus of continuity, forward difference operator},
url = {https://www.sciopen.com/article/10.3934/math.2026805},
doi = {10.3934/math.2026805},
abstract = {Let    f be a real-valued continuous function defined on the interval    [  0  ,  1  ], and let        B          n      ,      λ        (  f  ;  ⋅  ) be the associated    n-th degenerate Bernstein polynomial. We proved that, on any admissible interval,        |        B          n      ,      λ        (  f  ;  ⋅  )  −  f  (  ⋅  )      |   converges uniformly to    0 as    n  →  ∞. 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  ⁡  (  π  x  ) and its first-order derivative        f    ′    (  x  )  =  π  cos  ⁡  (  π  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        λ    n    →  0.}
}