We solve the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding the given polynomial into Bernstein polynomials. Then the coefficients of the expansion provide lower and upper bounds for the range and these bounds converge monotonically if the degree of the Bernstein polynomials is elevated. To obtain a faster improvement of the bou...

The paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so–called Bernstein–Bézier form of a polynomial.

One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the Weierstrass approximation theorem by S. Bernstein [2]: on the surface, this states that if f : [0, 1]...

When learning processes depend on samples but not on the order of the information in the sample, then the Bernoulli distribution is relevant and Bernstein polynomials enter into the analysis. We derive estimates of the approximation of the entropy function x log x that are sharper than the bounds from Voronovskaja’s theorem. In this way we get the correct asymptotics for the Kullback-Leibler di...

We study the existence and shape preserving properties of a generalized Bernstein operator Bn fixing a strictly positive function f0, and a second function f1 such that f1/f0 is strictly increasing, within the framework of extended Chebyshev spaces Un. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator Bn : C[a, b] → Un with strictly incr...

Given p polynomials of n variables over a field k of characteristic 0 and a point a ∈ k, we propose an algorithm computing the local Bernstein-Sato ideal at a. Moreover with the same algorithm we compute a constructible stratification of k such that the local Bernstein-Sato ideal is constant along each stratum. Finally, we present non-trivial examples computed with our algorithm.

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...

Let W : R → (0, 1] be continuous. Bernstein’s approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → ‖fW‖L∞(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950’s. Quantitative forms of the problem were actively investigated starting from the 1960’s. We survey old and recent aspects of this t...

We prove Bernstein type inequalities for algebraic polynomials on the finite interval I := [−1, 1] and for trigonometric polynomials on R when the roots of the polynomials are outside of a certain domain of the complex plane. The case of real vs. complex coefficients are handled separately. In case of trigonometric polynomials with real coefficients and root restriction, the Lpsituation will al...

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. analyse relations between the matrix coefficients of three term for orthonormal as well structure satisfied by these semiclassical polynomials, also a differential-difference equation is deduced. The extension Painlevé to case and two dimensional version Langmuir lattice are obtained.