in this paper, we introduce hybrid of block-pulse functions and bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. then, we utilize them to solvedelay differential equations and time-delay system. the method is based upon ex...

Iterated Bernstein polynomial approximations of degree n for continuous function which also use the values of the function at i/n, i = 0, 1, . . . , n, are proposed. The rate of convergence of the classic Bernstein polynomial approximations is significantly improved by the iterated Bernstein polynomial approximations without increasing the degree of the polynomials. The same idea applies to the...

This paper presents a numerical matrix method based on Bernstein polynomials (BPs) for approximate the solution of a system of m-th order nonlinear Volterra integro-differential equations under initial conditions. The approach is based on operational matrices of BPs. Using the collocation points,this approach reduces the systems of Volterra integro-differential equations associated with the giv...

*Correspondence: [email protected] Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, TR-07058, Turkey Abstract In this paper, we investigate some new identities related to the unification of the Bernstein-type polynomials, Bernoulli polynomials, Euler numbers and Stirling numbers of the second kind. We also give some remarks and applications of the Bernstein-ty...

Throughout his life Erdős showed a particular fascination with inequalities for constrained polynomials. One of his favorite type of polynomial inequalities was Markovand Bernstein-type inequalities. For Erdős, Markovand Bernstein-type inequalities had their own intrinsic interest. He liked to see what happened when the polynomials are restricted in certain ways. Markovand Bernstein-type inequa...

Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials.

The Bernstein-Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein-Bézier form. As application we consider Hermite interpolation with polynomials and splines.

We show that the classical Bernstein polynomials BN (f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN (f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler varie...