Lower Estimates for Centered Bernstein-Type Operators

We obtain converse inequalities of type A in the uniform distance for centered Bernstein-type operators in terms of suitable Ditzian—Totik moduli of smoothness. We use probabilistic representations of the operators in terms of stochastic processes as well as approximation-theoretic techniques. October 16, 2000. Date revised: January 15, 2001. Date accepted: March 12, 2001.     

Direct and Inverse Estimates for Bernstein Polynomials

Direct estimates for the Bernstein operator are presented by the Ditzian—Totik modulus of smoothness , whereby the step-weight φ is a function such that φ ² is concave. The inverse direction will be established for those step-weights φ for which φ ² and , are concave functions. This combines the classical estimate (φ=1 ) and the estimate developed by Ditzian and Totik ( ). In particular, the cases , λ∈[0,1] , are included. August 2, 1996. Date revised: March 28, 1997.     

Asymptotics of Differentiated Bernstein Polynomials

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C ^p . December 1, 1998. Dates revised: July 22, 1999 and January 11, 2000. Date accepted: February 1, 2000.     

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

This paper deals with the spectral problems for high-order nonsymmetric difference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters of these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonality, Hermite—Padé approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and inverse spectral problem of the operator. It is applied to the integration of hierarchy of the discrete KdV equations. The existence of a global solution is proved. July 13, 1998. Date revised: July 12, 1999. Date accepted: July 26, 1999.