Geometric series of positive linear operators and the inverse Voronovskaya theorem on a compact interval

Abstract

We define the associated geometric series for a large class of positive linear operators and study the convergence of the series in the case of sequences of admissible operators. We obtain an inverse Voronovskaya theorem and we apply our results to the Bernstein operators, the Bernstein-Durrmeyer-type operators, and the symmetrical version of Meyer-König and Zeller operators.2000 Mathematics Subject Classification: 41A36; 41A27; 41A35. Key words: positive linear operators, iterates of operators, series of operators, antiderivative, inverse Voronovskaya operators, Bernstein operators, Bernstein-Durrmeyer type operators, symmetrical version of Meyer-König and Zeller operators.

On Geometric Series of Positive Linear Operators

“…are well defined and the following result is true: Theorem A For any g ∈ C[0, 1] we have In [3], the geometric series are consider for a large class of operators, defined on an more extended space C ψ [0, 1] given by…”

“…The geometric series of operators were studied in [11], [1], [2], [3], [12]. The existence of these operators needs some restrictions of the domain of definition.…”

“…In [3], the geometric series are consider for a large class of operators, defined on an more extended space C ψ [0, 1] given by…”

Power series of the operators $$U_n^{\varrho }$$ U n ϱ

“…This article motivated a number of authors to study similar problems or give different proofs of Pȃltȃnea's main result. See [1], [2], [3], [11]. In one more and most significant article Pȃltȃnea [10] introduced a very interesting link between the classical Bernstein operators B n and the so-called "genuine Bernstein-Durrmeyer operators" U n , thus also bridging the gap between U n and piecewise linear interpolation in a most elegant way for the cases 0 < ̺ ≤ 1.…”