We discuss Beta operators with Jacobi weights on C[0, 1] for α, β ≥ −1, thus including the discussion of three limiting cases. Emphasis is on the moments and their asymptotic behavior. Extended Voronovskaya-type results and a discussion concerning the over-iteration of the operators are included.
We study power series of members of a class of positive linear operators reproducing linear function constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach. Keywords: Power series geometric series positive linear operator Bernsteintype operator genuine Bernstein-Durrmeyer operator degree of approximation eigenstructure moduli of continuity. MSC 2010: 41A10 41A17 41A25 41A36
The classical form of Grüss' inequality was first published by G. Grüss and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In the subsequent years, many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to the Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. The first and second moments of the operators will be of great interest.
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