0. Introduction; notation This paper lists the essential facts about the representation of polynomials in m variables as Bernstein polynomials. An expanded version may appear elsewhere. While univariate Bernstein polynomials are well studied see, e.g., Lorentz’ classical book Lorentz (1953), the multivariate version has only attracted attention sporadically. Lorentz’ book devotes just one page ...

Matrix methods for the computation of bounds range a complex polynomial and its modulus over rectangular region in plane are presented. The approach relies on expansion given into Bernstein polynomials. results extended to multivariate polynomials rational functions.

In this survey paper we give a short account on characterizations for very classical orthogonal polynomials via extremal problems and the corresponding inequalities. Beside the basic properties of the classical orthogonal polynomials we consider polynomial inequalities of Landau and Kolmogoroff type, some weighted polynomial inequalities in L2-norm of Markov-Bernstein type, as well as the corre...

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava–Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.

Composition of the Bernstein polynomials is an important research topic in computer-aided geometric design. This function is useful in implementing evaluation, subdivision, free-form deformation, trimming, conversion between tensor product and Bézier simplex forms, degree raising etc. To accomplish the composition, some numerically stable algorithms were introduced, such as blossoming algorithm...

Bernstein-Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.

which are now called Bernstein polynomials, in order to present a short proof of the Weierstrass Approximation Theorem. The subsequent history is well documented, see, e.g., [29] for the period up to 1955, the monograph [18] published in 1953, and the survey article [9] which appeared on the occasion of the hundredth anniversary of the above paper by Bernstein. Since the latter publication prov...

Using the language of Riordan arrays, we define a notion of generalized Bernstein polynomials which are defined as elements of certain Riordan arrays. We characterize the general elements of these arrays, and examine the Hankel transform of the row sums and the first columns of these arrays. We propose conditions under which these Hankel transforms possess the Somos-4 property. We use the gener...

Abstract. Since for q > 1, the q-Bernstein polynomials Bn,q are not positive linear operators on C[0, 1], the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. In this paper, new results on the approximation of continuous functions by the q-Bernstein polynomials in the case q > 1 are presented. It is shown that if f ∈ C[0, 1] and ...