Positive Linear Operators Research Articles

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.

Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

A Korovkin-type theorem for sequences of positive linear operators on function spaces

A Korovkin-type theorem for sequences of positive linear operators on function spaces

On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits

The equation Au=f with a linear symmetric positive definite operator A:D(A)⊂H→H having a discrete spectrum and dense image in a complex Hilbert space H is considered. This equation is transferred into the Hilbert space of finite orbits D(An) as well as into the Fréchet space of all orbits D(A∞), that is, the projective limit of the sequence of spaces {D(An)}. For an approximate solution of the inverse of A, linear spline central algorithms in these spaces are constructed. The convergence of the sequence of approximate solutions to the exact solution is proved. The obtained results are applied to the quantum harmonic oscillator operator Au(t)=−u″(t)+t2u(t), t∈R, in the Hilbert space of finite orbits D(An), and in the Fréchet space of all orbits D(A∞) that in this case coincides with the Schwartz space of rapidly decreasing functions. Some quantum mechanical interpretations of obtained results are also given.

$\mathcal{F-}$relative $\mathcal{A-}$summation process for double sequences and abstract Korovkin type theorems

In this paper, we first introduce the notions of $\mathcal{F-}$relative modular convergence and $\mathcal{F-}$relative strong convergence for double sequences of functions. Then we prove some Korovkin-type approximation theorems via $\mathcal{F-}$relative $\mathcal{A}-$summation process on modular spaces for double sequences of positive linear operators. Also, we present a non-trivial application such that our Korovkin-type approximation results in modular spaces are stronger than the classical ones and we present some estimates of rates of convergence for abstract Korovkin-type theorems. Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.

The Q-MBS Operators in a linear positive operator Voronovskaya style asymptotic formula

The paper reveals the development theory of modified Beta Stancu operators. Incorporate Q derivatives and get critical moments from another Q-Beta generator. There is also a simple module inference for Q operators. Primary moments, Voronovskaya method asymptotic process, and error calculations applying these characteristics are produced for these operators. We used Q integral properties to verify the recurrence theorem for Q analog moments of the updated Operators. We still have an asymptotic method. After the day, We examined all such Q operators’ updating to help estimates.

Voronovskaja-Type Quantitative Results for Differences of Positive Linear Operators

We consider positive linear operators having the same fundamental functions and different functionals in front of them. For differences involving such operators, we obtain Voronovskaja-type quantitative results. Applications illustrating the theoretical aspects are presented.

VORONOVSKAYA TYPE RESULTS AND OPERATORS FIXING TWO FUNCTIONS

The present paper deals with positive linear operators which fix two functions. The transfer of a given sequence (Ln) of positive linear operators to a new sequence (Kn) is investigated. A general procedure to construct sequences of positive linear operators fixing two functions which form an Extended Complete Chebyshev system is described. The Voronovskaya type formula corresponding to the new sequence which is strongly influenced by the nature of the fixed functions is obtained. In the last section our results are compared with other results existing in literature.

On approximation of two-variable function by linear positive polynomial operators

On approximation of two-variable function by linear positive polynomial operators.

Connections between the Approximation Orders of Positive Linear Operators and Their Max-Product Counterparts

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and Lp convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the Lp -norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.

Deferred Nörlund statistical relative uniform convergence and Korovkin-type approximation theorem

In this paper, we define the concept of statistical relative uniform convergence of the deferred Nörlund mean and we prove a general Korovkin-type approximation theorem by using this convergence method. As an application, we use classical Bernstein polynomials for defining an operator that satisfies our new approximation theorem but does not satisfy the theorem given before. Additionally, we estimate the rate of convergence of approximating positive linear operators by means of the modulus of continuity.

$f-$Statistical approximation to Bögel-type continuous functions

In this paper, considering the concept of $f-$statistical convergence which is a generalization of statistical convergence and is intermediate between the ordinary convergence and the statistical convergence, we obtain a $f-$statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued $B-$continuous functions on a compact subset of the real line. Furthermore, we compute the rates of $f-$statistical convergence with the help mixed modulus of smoothness.

Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion.

In this paper we propose an age-structured susceptible-infectious-susceptible epidemic model with nonlocal (convolution) diffusion to describe the geographic spread of infectious diseases via long-distance travel. We analyze the well-posedness of the model, investigate the existence and uniqueness of the nontrivial steady state corresponding to an endemic state, and study the local and global stability of this nontrivial steady state. Moreover, we discuss the asymptotic properties of the principal eigenvalue and nontrivial steady state with respect to the nonlocal diffusion rate. The analysis is carried out by using the theory of semigroups and the method of monotone and positive operators. The spectral radius of a positive linear operator associated to the solution flow of the model is identified as a threshold.

Kantorovich-type operators associated with a variant of Jain operators

"This note focuses on a sequence of linear positive operators of integral type in the sense of Kantorovich. The construction is based on a class of discrete operators representing a new variant of Jain operators. By our statements, we prove that the integral family turns out to be useful in approximating continuous signals de ned on unbounded intervals. The main tools in obtaining these results are moduli of smoothness of rst and second order, K-functional and Bohman- Korovkin criterion."

Sharp inequalities for the rates of convergence of the iterates of some operators which preserve the constants

"In this paper we give estimates for the rates of convergence for the iterates of some positive linear operators which preserve only the constants. We obtain sharp inequalities when we use both continuous functions and di eren- tiable functions. We present some optimal results for the Cesaro, Stancu and Schurer operators."

Certain approximation properties of Brenke polynomials using Jakimovski\u2013Leviatan operators

In this article, we establish the approximation by Durrmeyer type Jakimovski–Leviatan operators involving the Brenke type polynomials. The positive linear operators are constructed for the Brenke polynomials, and thus approximation properties for these polynomials are obtained. The order of convergence and the weighted approximation are also considered. Finally, the Voronovskaya type theorem is demonstrated for some particular case of these polynomials.

Approximation of Periodic Functions of Bounded Variation by Some Positive Linear Operators

We study the pointwise degree of approximation of periodic functions of bounded variation by positive linear polynomial operators. For the proof we used sub-additive majorants.

Bayesian nonparametrics for directional statistics

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature.From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.

On the convergence of sequences of positive linear operators and functionals on bounded function spaces

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on [ 0 , 1 ] [0,1] and on the d d -dimensional simplex and hypercube ( d ≥ 1 ) (d\geq 1) are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset K K of a locally convex space. As a consequence a general weak law of large numbers for sequences of K − K- valued random variables is derived.

On Markov Moment Problem and Related Results

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an Lν1 space, where ν is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.