Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) p...

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. much harder version under recent consideration to enforce bounds constraints on the polynomial. In this paper, we consider of constructioning such approximations using polynomials Bernstein basis. We a family inequality-constrained quadratic programs. univariate case, cone cons...

We construct Jacobi-weighted orthogonal polynomials (α,β,γ) n,r (u,v,w), α,β,γ > −1, α+ β + γ = 0, on the triangular domain T . We show that these polynomials (α,β,γ) n,r (u, v,w) over the triangular domain T satisfy the following properties: (α,β,γ) n,r (u,v,w) ∈ n, n≥ 1, r = 0,1, . . . ,n, and (α,β,γ) n,r (u,v,w) ⊥ (α,β,γ) n,s (u,v,w) for r =s. Hence, (α,β,γ) n,r (u,v,w), n= 0,1,2, . . ., r =...

This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial g, we can obtain a Bernstein polynomial of degree m with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0, 1 m , . . . , 1, provided that m is sufficiently large. The second result per...

We give a short and elementary proof of an inverse Bernsteintype inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle . The inequality is used to show that equally-spaced points solve a min-max-min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we s...

We combine recently-developed finite element algorithms based on Bernstein polynomials [1, 14] with the explicit basis construction of the finite element exterior calculus [5] to give a family of algorithms for the Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus base...

If is a polynomial of degree n such that in , then Govil [ Proc. Nat. Acad. Sci., Vol. 50, pp. 50-52, 1980. ] proved 
 provided and attain their maxima at the same point on circle where .
 Equality above inequality holds for.
 In this paper, we extend an improved version into polar derivative polynomial.