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On the Decomposition of Bernstein Operators

Let F n be the linear operators on C[0, 1] defined by , where B n are the classical Bernstein operators and are Beta operators. This decomposition of B n was investigated in Gonska et al. [6]. Although the operators F n are not positive, they have quite interesting properties. We obtain new results concerning the convergence of the sequence (F n ). An associated quadrature formula with positive coefficients and equidistant knots is investigated, as well as the eigenstructure of F n .