On multiplicativity of the Bernstein operator (original) (raw)

The Eigenstructure of the Bernstein Operator

Journal of Approximation Theory, 2000

The Bernstein operator B n reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of B n . Its eigenvalues are (n) k := n! (n ? k)! 1 n k ; k = 0; 1; : : :; n; and the corresponding monic eigenfunctions p (n) k are polynomials of degree k, which have k simple zeros in 0; 1]. By using an explicit formula, it is shown that p (n) k converges as n ! 1 to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p (n) k converge as n ! 1 to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at n + 1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein{Durrmeyer operator, which is self adjoint.

The complete asymptotic expansion for Bernstein operators

Journal of Mathematical Analysis and Applications, 2012

In this paper we study the asymptotic behavior of the classical Bernstein operators, applied to q-times continuously differentiable functions. Our main results extend the results of S.N. Bernstein and R.G. Mamedov for all q-odd natural numbers and thus generalize the theorem of E.V. Voronovskaja. The exact degree of approximation is also proved.

Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind

International Journal of Mathematics and Mathematical Sciences, 2009

Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form Cω 1 f; 1/ √ n with an unexplicit absolute constant C > 0 and the question of improving the order of approximation ω 1 f; 1/ √ n is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of ω 1 f; 1/ √ n and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions f including, for example, that of concave functions, we find the order of approximation ω 1 f; 1/n , which for many functions f is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.

Bernstein type operators with a better approximation for some functions

Applied Mathematics and Computation, 2013

V un f ðxÞ ¼ B n f ðu n ðxÞÞ ¼ X n k¼0 n k u n ðxÞ ð Þ k 1 À u n ðxÞ ð Þ nÀk f k n ; x 2 ½0; 1; ð2Þ where u n : ½0; 1 ! ½0; 1 are continuous functions satisfying u n ð0Þ ¼ 0 and u n ð1Þ ¼ 1. For u n ðxÞ ¼ x we get the classical Bernstein operator , and in general we have V un f ðxÞ ¼ B n f ðu n ðxÞÞ: Other operator s of the form (2) were considered in [