35 research outputs found
Gr\"uss and Gr\"uss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables
The first aim of this paper is to prove a Gr\"uss-Voronovskaya estimate for
Bernstein and for a class of Bernstein-Durrmeyer polynomials on . Then,
Gr\"uss and Gr\"uss-Voronovskaya estimates for their corresponding operators of
complex variable on compact disks are obtained. Finally, the results are
extended to Bernstein-Faber polynomials attached to compact sets in the complex
plane
Weighted Ostrowski-Gr\"uss type inequalities
Several inequalities of Ostrowski-Gruss-type availabe in the literature are
generalized by considering the weighted case of them. Involving the least
concave majorant of the modulus of continuity we provide upper error bounds of
such inequalities
Perturbed Bernstein-type operators
The present paper deals with modifications of Bernstein, Kantorovich,
Durrmeyer and genuine Bernstein-Durrmeyer operators. Some previous results are
improved in this study. Direct estimates for these operators by means of the
first and second modulus of continuity are given. Also the asymptotic formulas
for the new operators are proved
Classical Kantorovich operators revisited
The main object of this paper is to improve some of the known estimates for
classical Kantorovich operators. A quantitative Voronovskaya-type result in
terms of second moduli of continuity which improves some previous results is
obtained. In order to explain non-multiplicativity of the Kantorovich operators
a Chebyshev-Gr\"uss inequality is given. Two Gr\"uss-Voronovskaya theorems for
Kantorovich operators are considered as well
On Bullen's and related inequalities
The estimate in Bullen's inequality will be extended for continuous functions
using the second order modulus of smoothness. A different form of this
inequality will be given in terms of the least concave majorant. Also, the
composite case of Bullen's inequality is considered
S\"atze vom Bohman-Korovkin-Typ f\"ur lokalkonvexe Vektorverb\"ande
The convergence behavior of positive linear operators between certain locally
convev vector lattices is reconsidered here.Comment: in Germa
Lagrange-type operators associated with
We consider a class of positive linear operators which, among others,
constitute a link between the classical Bernstein operators and the genuine
Bernstein-Durrmeyer mappings. The focus is on their relation to certain
Lagrange-type interpolators associated to them, a well known feature in the
theory of Bernstein operators. Considerations concerning iterated Boolean sums
and the derivatives of the operator images are included. Our main tool is the
eigenstructure of the members of the class
Chebyshev-Gr\"{u}ss-type inequalities via discrete oscillations
The classical form of Gr\"uss' inequality, first published by G. Gr\"{u}ss in
1935, gives an estimate of the difference between the integral of the product
and the product of the integrals of two functions. In the subsequent years,
many variants of this inequality appeared in the literature. The aim of this
paper is to introduce a different approach, presenting a new
Chebyshev-Gr\"uss-type inequality and applying it to different well-known
linear, not necessarily positive, operators. Some conjectures are presented as
well. We also compare the new inequalities with some older results. This new
approach gives better estimates in some cases than the ones already known
Sur la suite des op\'erateurs Bernstein compos\'es
We consider a sequence of composite Bernstein operators and the quadrature
formulae associated with them. Upper bounds for the approximation error of
continuous functions and for the approximation of integrals of continuous
functions are given. The bounds are described in terms of moduli of continuity
of order one and two. Two inequalities of Tchebycheff-Gr\"uss-type are also
included.Comment: in Frenc
On the composition and decomposition of positive linear operators III: A non-trivial decomposition of the Bernstein operator
The central problem in this technical report is the question if the classical
Bernstein operator can be decomposed into nontrivial building blocks where one
of the factors is the genuine Beta operator introduced by M\"uhlbach and
Lupa\c{s}. We collect several properties of the Beta operator such as
injectivity, the eigenstructure and the images of the monomials under its
inverse. Moreover, we give a decomposition of the form where is a nonpositive linear operator
having quite interesting properties. We study the images of the monomials under
, its moments and various representations. Also an asymptotic formula of
Voronovskaya type for polynomials is given and a connection with a conjecture
of Cooper and Waldron is established. In an appendix numerous examples
illustrate the approximation behaviour of in comparison to