On the generalized Boolean sum Schurer-Stancu approximation formula

. In this paper, the Schurer-Stancu generalized Boolean sum (GBS, for short) approximation formula is considered and it’s remainder term is expressed in terms of bivariate divided differences. When the approximated function is sufficiently smooth, an upper bound estimation for the remainder term is also established. As particular cases, GBS Schurer and respectively GBS Bernstein approximation formulas are obtained and the expressions of their remainder are explicitly given.

The Bernstein quadrature formula revised

Starting with the Bernstein approximation formula on the interval [a, b] a corresponding composite quadrature formula is constructed. Its coefficients and an estimation for the remainder term are determined. Numerical examples are also presented.

Some approximation properties of a certain nonlinear Bernstein operators

The present paper concerns with a certain sequence of nonlinear Bernstein operators NBnf of the form (NBnf )(x) = ?nk=0 Pk,n (x,f (k/n)), 0 ? x ? 1, n ? N, acting on bounded functions on an interval [0, 1], where Pk, n satisfy some suitable assumptions. We will also investigate the pointwise convergence of this operators in some functional spaces. As a result, this study can be considered as an extension of the results dealing with the linear Bernstein Polynomials. As far as we know this kind of study is the first one on the nonlinear Bernstein approximation operators.