arXiv subjects
Publications and source records attributed to Heiner Gonska.
Extending an earlier estimate for the degree of approximation of overiterated univariate Bernstein operators towards the same operator of degree one, it is shown that an analogous result holds in the $d$-variate case. The method employed can be carried over to many other cases and is not restricted to Bernstein-type or similar methods.
We deal with multivariate Brass-Stancu-Kantorovich operators depending on a non-negative integer parameter and defined on the space of all Lebesgue integrable functions on a unit hypercube. We prove $L^{p}$-approximation and provide estimates for the $L^{p}$-norm of the error of approximation in terms of a multivariate averaged modulus of continuity and of the corresponding $L^{p}$-modulus.
The Bernstein-type operator of Aldaz, Kounchev and Render (2009) is discussed. New direct results in terms of the classical second order modulus as well as in a modification following Marsden and Schoenberg are given.
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-Grüss inequality is given. Two Grüss-Voronovskaya theorems for Kantorovich operators are considered as well.
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.
The classical form of Grüss' inequality was first published by G. Grüss in 1935 and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. After that many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. First and second moments of the operators are used to explain the situation.
We consider a generalized form of certain integral inequalities given by Guessab, Schmeisser and Alomari. The trapezoidal, mid point, Simpson, Newton-Simpson rules are obtained as special cases. Also, inequalities for the generalized Alomari functional in terms of the $n$-th order modulus, $n=\overline{1,4}$, are given and applied to some known quadrature rules.
The convergence behavior of positive linear operators between certain locally convev vector lattices is reconsidered here.
We consider a sequence of composite bivariate Bernstein operators and the cubature formula associated with them. The upper bounds for the remainder term of the cubature formula are described in terms of moduli of continuity of order two. Also we include some results showing how non-multiplicative the integration functional is.
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.
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.
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.
We discuss Beta operators with Jacobi weights on $C[0,1]$ for $α,β\geq-1$, thus including the discussion of three limiting cases. Emphasis is on the moments and their asymptotic behavior. Extended Voronovskaya-type results and a discussion concerning the over-iteration of the operators are included.
We study power series of members of a class of positive linear operators reproducing linear function and constituting a link between genuine Bernstein-Durrmeyer and classical Bernstein operators. Using the eigenstructure of the operators we give a non-quantitative convergence result towards the inverse Voronovskaya operators. We include a quantitative statement via a smoothing approach.
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-Grüss-type are also included.
The classical form of Grüss' inequality, first published by G. Grüss 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-Grüss-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.
The first aim of this paper is to prove a Grüss-Voronovskaya estimate for Bernstein and for a class of Bernstein-Durrmeyer polynomials on $[0, 1]$. Then, Grüss and Grüss-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.
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 Mühlbach and Lupaş. 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 $B_n = \bar{\mathbb{B}}_n \circ F_n $ where $F_n$ is a nonpositive linear operator having quite interesting properties. We study the images of the monomials under $F_n$, 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 $F_n$ in comparison to $B_n$.