Bernstein polynomials

Bernstein Polynomials for a Standard Module Function on the Symmetric Interval

Authors: Tikhonov I. V., Sherstyukov V. B., Petrosova M. A.

Bernstein polynomials are studied on a symmetric interval. Basic relations connected with Bernstein polynomials for a standard module function are received. By the Templ’s formula we establish recurrence relations from which the Popoviciu’s expansion is derived. Suitable formulas for the first and second derivatives are found. As a result an explicit algebraic form for Bernstein polynomials is obtained. We also notice some corollaries. 

On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces

Authors: Shakh-Emirov T. N.

Let E = [0, 1] and let a function p(x) > 1 be measurable and essentially bounded on E. We denote by L p(x) (E) the set of measurable function f on E for which R E |f(x)| p(x) dx < ∞. The convergence of a sequence of operators of Bernstein – Kantorovich {Kn(f, x)} ∞n=1 to the function f in Lebesgue spaces with variable exponent L p(x) (E) is studied.

Gluing Rule for Bernstein Polynomials on the Symmetric Interval

Authors: Tikhonov I. V., Sherstyukov V. B., Petrosova M. A.

We study special laws that arise in a sequence of the Bernstein polynomials on a symmetric interval. In particular, we set the exact rule of regular pairwise coincidence (gluing rule) which is acting for the Bernstein polynomials of a piecewise linear generating function with rational abscissas of break points. The accuracy of this rule for convex piecewise linear generating functions is shown. The possibility of “random” gluing for the Bernstein polynomials in a non-convex case is noted. We give also some examples and
illustrations.