Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions

Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

“…The inequality of Theorem 3.3 can also be derived from Theorem 5.2, formula (5.3), and case r D 0 in [1]. For a similar inequality, see Theorem 2.3 and inequality (2.11) in [2].…”

“…In the present paper, we continue our research on a class of one-parameter operators U n of BernsteinDurrmeyer type that preserve linear functions and constitute a link between the so-called "genuine BernsteinDurrmeyer operators" U n and the classic Bernstein operators B n : A predecessor of this paper (see [1]) will appear soon in the Czechoslovak Mathematical Journal. The investigation on the operators in question started in a 2007 note by the second author (see [2]).…”

“…For several further properties of U n ; the reader is referred to our recent article [1], some results of which are also used in the present note.…”

Approximation by Bernstein–Durrmeyer-type operators in compact disks

“…In [3] a Voronovskaja-type result with a quantitative estimate for complex Bernstein polynomials in compact disks was obtained. In [4][5][6][7][8][9][10][11]1,12,13] similar results for Bernstein-Stancu, Kantorovich-Stancu and q-Sancu polynomials were obtained while in [14] similar results for Bernstein-Schurer polynomials were proved. Very recently, Anastassiou and Gal [15], Gal [7] studied the order of simultaneous approximation and Voronovskaja-kind results with quantitative estimates for complex Bernstein-Durrmeyer and genuine Durrmeyer polynomials attached to analytic functions on compact disks.…”

“…Among many articles written on the genuine Bernstein-Durrmeyer operators, we mention here only the ones by Gonska et al [9], Parvanov and Popov [18], Sauer [19], Waldron [20], Mahmudov and Sabancigil [12] and the book of Păltănea [21]. Recently, Păltănea [22], Gonska and Păltănea [10,11] introduced and studied a one-parameter class of positive linear operators constituting a non-trivial link between the Bernstein and the genuine Bernstein-Durrmeyer operators.…”