Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

Abstract

The aim of this paper is the study of a new sequence of positive linear approximation operators M n, * on C([0, 1]) which generalize the classical Bernstein Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators. Academic Press

Durrmeyer type operators on a simplex

“…The eigen-structure of this operators was studied in [22]. For other modifications of Bernstein-Durrmeyer operators mention [18], [27], [16], [3], [1], [15].…”

Overiterated Linear Operators and Asymptotic Behaviour of Semigroups

“…(ii) (A, D(A)) is closable and the closure (Ā, D(Ā)) is the infinitesimal generator of a C 0 -semigroup (T (t)) t≥0 on X; The approximation of semigroups as in (iii) and (iv), with X a function space and (L n ) n≥1 a sequence of positive linear operators, has been intensively investigated by Altomare and his school: see, e.g., [1], [2], [3] and [4] and the references quoted therein, where many concrete examples illustrating the above setting may be found.…”

APPROXIMATION BY POSITIVE OPERATORS OF THE C0–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART II

“…The results obtained for compact intervals would be compared, among others, with the ones of [9][10][11][12][13], and [17], where other representing sequences of positive linear operators have been introduced in the same spirit of this paper.…”

“…We finally refer to [10], [12], [13] and [17], where other particular cases of the differential operator (3.20) have been studied in the same spirit of this paper and where other representing sequences of positive operators have been introduced. 3.…”