Transformation of the second order modulus by positive linear operators
Abstract We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.
Estimates with Second Order Moduli
In this chapter we continue the study of estimating the degree of an approximation using general linear positive operators by considering combinations of first and second order moduli, in terms of the moments of order 0, 1, and 2, see Remark 1.2.4. Estimates with such combinations of first and second order modulus, (and also with the absolute value of the function, which can be regarded as a modulus of order 0) are more refined then estimates using only the first modulus. A first observation is that, from estimates with the second order modules, one can derive estimates with the first order modulus. A second observation is the fact that such combinations decompose the error of approximation in three components, corresponding to three specific features of the functions that affect the error: amplitude, deviation from the linear functions, and deviation from the polynomials of degree 2. Roughly speaking, these moduli measure the deviation from the test functions of the algebraic Chebychev system.
Weighted approximation by multidimensional Bernstein operators
Abstract The purpose of this paper is to give inverse theorems for univariate and multivariate Bernstein operators with some Jacobi weights. A crucial tool in our approach is a decomposition technique which is especially useful for dealing with interpolation theorems.