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...

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subd...

In the present paper a Korovkin-type theorem is proposed for the approximation operators defined by the inverse Ftransforms. These results allow us to choose between a variety of shapes to be used as atoms of the fuzzy partitions used within the F-transform’s framework. In this way we can enlarge considerably the class of F-transforms proposed recently by I. Perfilieva. The new fuzzy partitions...

Let Bn(f, q; x), n = 1, 2, . . . be the q-Bernstein polynomials of a function f ∈ C[0, 1]. In the case 0<q < 1, a sequence {Bn(f, q; x)} generates a positive linear operator B∞ = B∞,q on C[0, 1], which is called the limit q-Bernstein operator. In this paper, a connection between the smoothness of a function f and the analytic properties of its image under B∞ is studied. © 2005 Elsevier Inc. All...

A simple method is presented by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials into Bernstein polynomials.

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We study inĄnite exchangeability assessments for them, and give a counterpart of de FinettiŠs inĄnite representation theorem. We show how the inĄnite representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bar...

We survey some recent applications of Bernstein expansion to robust stability, viz. checking robust Hurwitz and Schur stability of polynomials with polynomial parameter dependency by testing determinantal criteria and by inspection of the value set. Then we show how Bernstein expansion can be used to solve systems of strict polynomial inequalities.

It is well known that the degree-raised Bernstein-B ezier coeecients of degree n of a polynomial g converge to g at the rate 1=n. In this paper we consider the polynomial An(g) of degree n interpolating the coeecients. We show how An can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives An(g) (r) converge uniformly to g (r) at the rate 1=n for all r. We also ...

We investigate the iterates of positive linear operators on C[0, 1] which preserve the linear functions, and the asymptotic behaviour of the associated semigroup. The general results are applied to the Bernstein-Schnabl operators on the unit interval.

We introduce the local derivatives of a Weyl algebra and prove a theorem of I. N. Bernstein concerning the existence of certain polynomials relating to the action of local derivatives.