In 1987, C. Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this article is to give a more elementary and constructive proof of the result of C. Sabbah based on the notion of the analytic Gröbner fan of a D-module. This paper is a translation of [Bah]. Introduction and statement of the main results Fix two integers n ≥ 1 and p...

We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators. © 2006 Elsevier Inc. All rights reserved. MSC: 41A36; 26D15

Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K := R (mod 2π), on [−1, 1] and on R. The key result is the establishment of the inequality

This paper deals with the control of polynomial nonlinear systems which are affine in the control. We use Bernstein polynomials and Polytopic Linear Differential Inclusion (PLDI) to design gain-scheduled controllers using a guaranteed cost framework.

In this paper, when approximating a continuos non-negative function on the unit interval, we present an alternative way to the classical Bernstein polynomials. Our new operators become nonlinear, however, for some classes of functions, they provide better error estimations than the Bernstein polynomials. Furthermore, we obtain a simultaneous approximation result for these operators. 2010 Mathem...

An accurate method is proposed to solve problems such as identification, analysis and optimal control using the Bernstein orthonormal polynomials operational matrix of integration. The Bernstein polynomials are first orthogonalized, normalized and then their operational matrix of integration is obtained. An example is given to illustrate the proposed method. Mathematics Subject Classification: ...

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. In this paper a special class of rational Brzier curves is presented that correspond to low-harmonic trajectory patterns. Harmonic Bernstein polynomials and harmonic deCasteljau algorithm are ...

This paper describes an algorithm that can be used to effectively solve polynomial constraints over finite domains. Such constraints are expressed in terms of inequalities of polynomials with integer coefficients whose variables are assumed to be defined over proper finite domains. The proposed algorithm first reduces each constraint to a canonical form, i.e., a specific form of inequality, the...

We present a procedure for computing the coeecients of the expansion of a bivariate polynomial into Bernstein polynomials over subtriangles. These triangles are generated by partitioning the standard simplex of IR 2. The coeecients are computed directly from the coeecients on the subdivided triangle from the preceding subdivision level. This allows a recursive computation of the coeecients and ...