We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functi...

Interval Newton methods are widely used to find reliable enclosures for the roots of the polynomial systems. But the computation process needs the evaluation of an enclosure for the derivative of the polynomial and evaluation of the polynomial at a particular point. Again, the subdivisions (if any are needed) will require repeated evaluation of the function values. We propose an alternative app...

Let Bm(f) be the Bernstein polynomial of degree m. The Generalized Bernstein polynomials

We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two cl...

Abstract. We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein–Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the deriva...

One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leis...

in this paper we present a new method for solving fractional optimal control problems with delays in state and control. this method is based upon bernstein polynomial basis and using feedback control. the main advantage of using feedback or closed-loop controls is that they can monitor their effect on the system and modify the output accordingly. in this work, we use bernstein polynomials to tr...

In the present note, we introduce a Bezier variant of a new type of Bernstein Durrmeyer operator, which was introduced by Gupta [3]. We estimate the rate of convergence by using the decomposition technique of functions of bounded variation and applying the optimum bound. It is observed that the analysis for our Bezier variant of new Bernstein Durrmeyer operators is different from the usual Bern...