102 research outputs found
Approximation by Genuine -Bernstein-Durrmeyer Polynomials in Compact Disks in the case
This paper deals with approximating properties of the newly defined
-generalization of the genuine Bernstein-Durrmeyer polynomials in the case
, whcih are no longer positive linear operators on . Quantitative
estimates of the convergence, the Voronovskaja type theorem and saturation of
convergence for complex genuine -Bernstein-Durrmeyer polynomials attached to
analytic functions in compact disks are given. In particular, it is proved that
for functions analytic in \left\{ z\in\mathbb{C}:\left\vert z\right\vert
q, the rate of approximation by the genuine
-Bernstein-Durrmeyer polynomials () is of order versus
for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit
formulas of Voronovskaja type for the genuine -Bernstein-Durrmeyer for
Simultaneous approximation by operators of exponential type
There are many results on the simultaneous approximation by sequences of
special positive linear operators. In the year 1978, Ismail and May as well as
Volkov independently studied operators of exponential type covering the most
classical approximation operators. In this paper we study asymptotic properties
of these class of operators. We prove that under certain conditions, asymptotic
expansions for sequences of operators belonging to a slightly larger class of
operators, can be differentiated term-by-term. This general theorem contains
several results which were previously obtained by several authors for concrete
operators. One corollary states, that the complete asymptotic expansion for the
Bernstein polynomials can be differentiated term-by-term. This implies a
well-known result on the Voronovskaja formula obtained by Floater
Approximation by q-Szasz operators
his paper deals with approximating properties of the newly defined
-generalization of the Sz\'{a}sz operators in the case . Quantitative
estimates of the convergence in the polynomial weighted spaces and the
Voronovskaja's theorem are given. In particular, it is proved that the rate of
approximation by the -Sz\'{a}sz operators () is of order
versus for the classical Sz\'{a}sz--Mirakjan operators
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