Bernstein-type Operators Research Articles

Iterated Boolean Sums of Bernstein Type Operators

The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples.

PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$, θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

Generalized Bernstein type operators

In this paper we investigate certain properties of a class of generalized Bernstein type operators

Iterates of a modified Bernstein type operator

Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of some modified Bernstein type operators.

Intermediate Voronovskaja type results for the Lototsky–Bernstein type operators

We prove intermediate Voronovskaja type results for the Lototsky–Bernstein type operators. Various concrete applications are given.

Quantitative results for positive linear operators which preserve certain functions

Abstract In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented.

Pointwise convergence of the Bernstein–Durrmeyer operators with respect to a collection of measures

In this paper we consider a generalization of the Bernstein–Durrmeyer operator where the integrals are taken with respect to measures that may vary from term to term. This construction is more general than the one considered by the first named author and her coauthors earlier, and it includes a number of well-known operators of Bernstein type as particular cases. We give conditions on the collections of measures that guarantee pointwise convergence at a point of continuity of a function.

Blending type approximation by bivariate generalized Bernstein type operators

In this article we establish an extension of the bivariate generalization of the Bernstein type operators involving parameters. For these operators we obtain a Voronovskaja type and Grüss Voronovskaja type theorems and the degree of approximation by means of the Lipschitz type space. Further, we present the associated Generalized Boolean Sum (GBS) operators and establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MAPLE software.

On approximation by some Bernstein–Kantorovich exponential-type polynomials

Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and Neamtu (Adv Comput Math 12:133–149, 2000). They proposed an analog of classical Bernstein operator and proved some convergence results for continuous functions. Herein, we introduce their integral extensions in Kantorovich sense by replacing the usual differential and integral operators with their more general analogues. By means of these operators, we are able to reconstruct the functions which are not necessarily continuous. It is shown that the operators form an approximation process in both $$C\left[ 0,1\right] $$ and $$L_{p,\mu }\left[ 0,1\right] $$ , which is an exponentially weighted space. Also, quantitative results are stated in terms of appropriate moduli of smoothness and K-functionals. Furthermore, a quantitative Voronovskaya type result is presented.

On pointwise approximation properties of certain nonlinear Bernstein operators

The present study is concerned with the nonlinear Bernstein type operators $NB_{n}f$, acting on bounded functions, where the kernel function of the operators provide some convenient assumptions. Especially, some pointwise convergence results for these type operators are achieved at a generalized Lebesgue point of the function $f$.

Generalized Bernstein type operators on unbounded interval and some approximation properties

In the present paper, we construct a new family of Bernstein type operators on infinite interval by using exponential function ax. We study some approximation results for these new operators on the interval [0,1).

Degree of Approximation for Bivariate Generalized Bernstein Type Operators

In this paper we study an extension of the bivariate generalized Bernstein operators based on a non-negative real parameters. For these operators we obtain the order of approximation using Peetre’s K-functional, a Voronovskaja type theorem and the degree of approximation by means of the Lipschitz class. Further, we consider the Generalized Boolean Sum operators of generalized Bernstein type and we study the degree of approximation in terms of the mixed modulus of continuity. Finally, we show the comparisons by some illustrative graphics in Maple for the convergence of the operators to certain functions.

Nested sequences of rational spaces: Bernstein approximation, dimension elevation, and Pólya-type theorems on positive polynomials

On a given closed bounded interval, an infinite nested sequence of Extended Chebyshev spaces containing the constants automatically generates an infinite sequence of positive linear operators of Bernstein-type. Unlike the polynomial framework, this situation does not guarantee convergence of the corresponding approximation process. Obviously, convergence cannot be obtained without the density of the union of all the involved spaces in the set of continuous functions equipped with the uniform norm. The initial purpose of this work was to answer the following question: conversely, is density sufficient to guarantee convergence? Addressing this issue is all the more natural as density was indeed proved to imply convergence in the special case of nested sequences of Müntz spaces on positive intervals. In this paper we give a negative answer to the aforementioned question by considering nested sequences of rational spaces defined by infinite sequences of real poles outside the given interval. Surprisingly, in this rational context, we show that ensuring convergence is equivalent to determining all Pólya positive sequences, in the sense of all infinite sequences of positive numbers which guarantee Pólya-type results for the positivity of univariate polynomials on the non-negative axis. This interesting connection with Pólya positive sequences enables us to produce a simple necessary and sufficient condition for the poles to ensure convergence, thanks to results by Baker and Handelman on strongly positive sequences of polynomials.

Applications of the Shorgin Identity to Bernstein Type Operators

By establishing an identity between a sequence of Bernstein-type operators and a sequence of Szasz–Mirakyan operators, we prove that the convergence of Bernstein-type operators is related to convergence with respect to Szasz–Mirakyan operators. As one application of this identity, we prove that whenever the parameters are conveniently chosen, if $$f\in C[0,\infty )$$ satisfies a growth condition of the form $$|f(t)|\le C e^{\alpha t}(C,\alpha \in \mathbb {R}^+)$$ , then the classical Bernstein operators $$B_{mn}(f(nu),x/n)$$ converge to the Szasz–Mirakyan operator $$S_m(f,x)$$ . This convergence generalizes the classical result of De la Cal and Liquin to unbounded functions; moreover, the rth derivative of $$B_{mn}(f(nu),x/n)$$ converges to the rth derivative of $$S_m(f,x)$$ . As another application of this identity, we derive Voronowskaja type result for the general Lototsky–Bernstein operators.

Rate of convergence of q-analogue of a class of new Bernstein type operators

Rate of convergence of q-analogue of a class of new Bernstein type operators

Bernstein-type operators that reproduce exponential functions

Bernstein-type operators that reproduce exponential functions

A note on the fixed points of positive linear operators

We prove that a property concerning the existence of the fixed points of the Bernstein-type operators remains valid for a large class of positive linear operators. More exactly, we prove that there is no positive linear analytic operator L:C[0,1]→C[0,1] possessing as fixed points two nonconstant monomials xi.

A basic problem of (p,q)-Bernstein-type operators

In this note, we give an elaboration of a basic problem on convergence theorem of (p, q)-analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of (p_{n},q_{n})-integer satisfying lim _{ntoinfty }[n]_{p_{n},q_{n}}=infty with lim _{ntoinfty}p_{n}=1 and lim _{ntoinfty}q_{n}=1 under 0< q_{n}< p_{n}leq1. Our results provide an erratum to corresponding results on (p,q)-analogue of Bernstein-type operators that appeared in recent literature.

Feller’s Scheme in Approximation by Nonlinear Possibilistic Integral Operators

ABSTRACTBy analogy with Feller’s general probabilistic scheme used in the construction of many classical convergent sequences of linear operators, in this paper, we consider a Feller-kind scheme based on the possibilistic integral, for the construction of convergent sequences of nonlinear operators. In particular, in the discrete case, all the so-called max-product Bernstein-type operators and their qualitative convergence properties are recovered. Also, discrete nonperiodic nonlinear possibilistic convergent operators of Picard type, Gauss–Weierstrass type and Poisson–Cauchy type are studied and the possibility of introduction of discrete periodic(trigonometric) nonlinear possibilistic operators of de la Vallée–Poussin type, of Fejér type and of Jackson type is mentioned as future directions of research.

A Proof of a Conjecture About the Asymptotic Formula of a Bernstein Type Operator

In this paper we validate a conjecture about the asymptotic behavior of a Bernstein type operator.