TY  - JOUR
AU  - Khalil, Murad
AU  - Park, Seongho
AU  - Lee, Si Hyeon
AU  - Hwang, Kyo-Shin
PY  - 2026
TI  - Uniform approximation for degenerate Bernstein polynomials
JO  - AIMS Mathematics
SP  - 19824
EP  - 19854
VL  - 11
IS  - 7
AB  - 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.
UR  - https://doi.org/10.3934/math.2026805
DO  - 10.3934/math.2026805