On the Approximation by Linear Positive Operators

Yet another look at positive linear operators, q-monotonicity and applications

“…The operators M n are particular instances (for the generating polynomials constructed in Lemma 2.13) of, what we call, Gavrea's operators H n whose construction is based on Ioan Gavrea's clever combination of genuine Bernstein-Durrmeyer polynomials with coefficients of appropriate generating polynomials (see (2.15)). This construction heavily relies on a very powerful but little known and hardly accessible article by Alexandru Lupaş [15], extending the Bernstein-Durrmeyer operators by introducing ultraspherical weights (see Section 3 for details).…”

“…The main results in this paper (as well as all results from [4] and [9] that we need) greatly depend on (in our opinion, a rather interesting) paper by A. Lupaş [15] which does not seem to be readily available. Hence, in this section, we state and provide alternative elementary proofs for all theorems from [15] that we use. For α > −1, let…”

“…Alternatively, (3.8) can be proved by induction using the recurrence equation (3.2). Yet another way to prove (3.8) is to use several results from the theory of hypergeometric functions as was originally done by Lupaş in [15].…”

On new class of linear and positive operators

“…It is easy to see that for β = 0, the operators (3.2) reduces to Lupaş operator [1]. Along the paper, when we will deal with approximation results, the parameter β will be assumed to be a sequence β n which tends to zero as n → +∞.…”

“…In the year 1995, Lupaş [1] proposed to study the following sequence of linear and positive operators: This L n operators has a form very similar with the Szász-Mirakyan operators. In [2] Agratini modified the operators (1.1) into integral form in Kantorovich sense and established their approximation properties.…”