Positive Linear Operators Research Articles

Reflexivity for spaces of regular operators on Banach lattices

We prove that if Banach lattices E E and F F are reflexive and each positive linear operator from E E to F F is compact then L r ( E ; F ) {\mathcal L}^r(E;F) , the space of all regular linear operators from E E to F F , is reflexive. Conversely, if E ∗ E^\ast or F F has the bounded regular approximation property then the reflexivity of L r ( E ; F ) {\mathcal L}^r(E;F) implies that each positive linear operator from E E to F F is compact. Analogously we also study the reflexivity for the space of regular multilinear operators on Banach lattices.

Approximation of Real Functions by a Generalization of Ismail–May Operator

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth.

A Voronovskaja type formula for Soardi’s operators

In 1991 Soardi introduced a sequence of positive linear operators beta _{n} associating to each function fin Cleft[ 0,1right] a polynomial function which is closely related to the Bernstein polynomials on left[ -1,+1right] . One of the authors already studied the operators beta _{n} in several papers. This paper is devoted to other properties of Soardi’s operators. We introduce a version {tilde{beta }}_{n} which can be expressed in terms of the classical Bernstein operators and present the relations between beta _{n} and {tilde{beta }}_{n}. We derive Voronovskaja-type results for both beta _{n} and {tilde{beta }}_{n}. Furthermore, rates of convergence for {tilde{beta }}_{n}, respectively beta _{n}, are estimated. Finally, we study the first and second moments of beta _{n}.

The rate of approximation of functions in an infinite interval by positive linear operators

Abstract An estimation of the approximation rate by positive linear operators of functions defined on the positive half line that have finite limit at infinity is discussed. For the application of the obtained results, we introduce the positive linear operators which preserve 2 a ⁢ x 2^{ax} , a > 0 a>0 , and obtain their uniform convergence, order of approximation via a certain weighted modulus of continuity, and a quantitative Voronovskaya type theorem.

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Investigating (p,q)-hybrid Durrmeyer-type Operators in terms of Their Approximation Properties

This study introduces (p,q)-hybrid Durrmeyer-Stancu type linear positive operators, which are generalized forms of q-hybrid Durrmeyer-Stancu-type linear positive operators and examines their approximation properties. The first modulus of continuity on a finite interval is introduced using Peetre’s K-functional. Then, the weighted approximation theorem in a weighted space is provided using Gadzhiev’s weighted Korovkin-type theorem. Finally, these operators’ rates of convergence are obtained for the continuous functions.

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.

A Note of Jessen’s Inequality and Their Applications to Mean-Operators

A variant of Jessen’s type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra, is obtained. The corresponding mean value theorems lead to a new family of mean-operators.

Voronovskaya-type Results for Positive Linear Operators of Exponential Type and their Derivatives

Voronovskaya-type Results for Positive Linear Operators of Exponential Type and their Derivatives

Lupaş Bernstein-Kantorovich operators using Jackson and Riemann type (p,q)-integrals

In this paper, Lupa? Bernstein-Kantorovich operators have been studied using Jackson and Riemann type (p,q)-integrals. It has been shown that (p, q)-integrals as well as Riemann type (p, q)-integrals are not well defined for 0 < q < p < 1 and thus further analysis is needed. Throughout the paper, the case 1 ? q < p < ? has been used. Advantages of using Riemann type (p, q)-integrals are discussed over general (p, q)-integrals. Lupa? Bernstein-Kantorovich operators constructed via Jackson integral need not be positive for every f ? 0. So to make these operators based on general (p, q)-integral positive, one need to consider strictly monotonically increasing functions, and to handle this situation Lupa? Bernstein-Kantorovich operators are constructed using Riemann type (p, q)-integrals. However Lupa? (p, q)-Bernstein-Kantorovich operators based on Riemann type (p, q)-integrals are always positive linear operators. Approximation properties for these operators based on Korovkin?s type approximation theorem are investigated. The rate of convergence via modulus of continuity and function f belonging to the Lipschitz class is computed.

Dunkl analogue of Sz$ \acute{a} $sz-Schurer-Beta operators and their approximation behaviour

<p style='text-indent:20px;'>The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Sz<inline-formula><tex-math id="M2">\begin{document}$ \acute{a} $\end{document}</tex-math></inline-formula>sz-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces.</p>

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C^* -algebra \mathcal{A} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) as a linear combination of completely positive maps from \mathcal{A} to \mathcal{L}(\mathcal{Y}) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map \phi from an operator system \mathbb{S} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) can be extended to a completely positive map \phi_e from a C^* -algebra containing \mathbb{S} to \mathcal{L}(\mathcal{Y})) , and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).

A Bernstein-Schnabl type operator: Applications to difference equations

We consider a sequence of positive linear operators Ln of Bernstein-Schnabl type. It was studied in the literature from various points of view; we provide new properties of it. The eigenstructure of these operators is described. We investigate the kernel of Ln which is related with the set of solutions of a difference equation. Several algorithms are proposed in order to solve the involved problems.

Optimal estimates of approximation errors for strongly positive linear operators on convex polytopes

In the present investigation, we introduce and study linear operators, which underestimate every strongly convex function. We call them, for brevity, sp-linear (approximation) operators. We will provide their sharp approximation errors. We show that the latter is bounded by the error approximation of the quadratic function. We use the centroidel Voronoi tessellations as a domain partition to construct best sp-linear operators. Finally, numerical examples are presented to illustrate the proposed method.

On Shape Parameter α‐Based Approximation Properties and q‐Statistical Convergence of Baskakov‐Gamma Operators

We construct a novel family of summation‐integral‐type hybrid operators in terms of shape parameter α ∈ [0,1] in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre’s K‐functional, Lipschitz class, and second‐order modulus of smoothness. The approximation results are then obtained in weighted space. Finally, for these operators q‐statistical convergence is also investigated.

GENERALIZED MULTIVARIATE PRABHAKAR TYPE FRACTIONAL INTEGRALS AND INEQUALITIES

We introduce here the mixed generalized multivariate Prabhakar type left and right fractional integrals and study their basic properties, such as preservation of continuity and their boundedness as positive linear operators. Then we produce an interesting variety of related multivariate left and right fractional Hardy type inequalities under convexity. We introduce also other related multivariate fractional integrals

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function τ by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves {1,τ,τ2} instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results.

An Inequality Involving Some Linear Positive Operators and Box-Convex Functions

An Inequality Involving Some Linear Positive Operators and Box-Convex Functions

Further results on A-numerical radius inequalities

Let A be a bounded linear positive operator on a complex Hilbert space {mathcal {H}}. Furthermore, let {mathcal {B}}_Amathcal {(H)} denote the set of all bounded linear operators on {mathcal {H}} whose A-adjoint exists, and {mathbb {A}} signify a diagonal operator matrix with diagonal entries are A. Very recently, several {mathbb {A}}-numerical radius inequalities of 2times 2 operator matrices were established. In this paper, we prove a few new {mathbb {A}}-numerical radius inequalities for 2times 2 and ntimes n operator matrices. We also provide a new proof of an existing result by relaxing a sufficient condition “A is strictly positive”. Our proofs show the importance of the theory of the Moore–Penrose inverse of a bounded linear operator in this field of study.

Approximation by mixed positive linear operators based on second-kind beta transform

In this paper, we consider mixed approximation operators based on second-kind beta transform using Szász–Mirakjan operators. For the proposed operators, we establish some direct results, Voronovskaya-type theorem, quantitative Voronovskaya-type theorem, Grüss–Voronovskaya-type theorem, weighted approximation and functions of bounded variation.