Positive Linear Operators Research Articles

Composition of some positive linear integral operators

Abstract This article is devoted to constructing sequences of integral operators with the same Voronovskaja formula as the generalized Baskakov operators, but having different behavior in other respects. For them, we investigate the eigenstructure, the inverses, and the corresponding Voronovskaja type formulas. A general result of Voronovskaja type for composition of operators is given and applied to the new operators. The asymptotic behavior of differences between the operators is investigated, and as an application, we obtain a formula involving Euler’s gamma function.

On Positive Linear Operators linking Gamma, Mittag-Leffler and Wright Functions

On Positive Linear Operators linking Gamma, Mittag-Leffler and Wright Functions

On Some Positive Linear Operators Preserving the B^{-1}-Convexity of Functions

In the study, we first give an inequality that non-negative differentiable functions must satisfy to be B^{-1}-convex. Then, using the inequality, we show that the B^{-1}-convexity property of functions is preserved by Bernstein-Stancu operators, Sz'asz-Mirakjan operators and Baskakov operators. In addition, we compare the concepts B^{-1}-convexity and B-concavity of functions.

On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.

The Convergence of Some Positive Linear Operators on the Space of Multivariate Continuous Periodic Functions

As a consequence of a general result, we prove that in the case of singular integrals the set of convergence consists only of the two functions 1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{1}$$\\end{document} and cos\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cos $$\\end{document}. We prove also a multivariate version of this result and apply it to find the necessary and sufficient conditions for the convergence of the sequences of positive linear operators associated to the rectangular and triangular summation.

On the convergence of sequences of positive linear operators towards composition operators

On the convergence of sequences of positive linear operators towards composition operators

Approximation theorems via Pp -statistical convergence on weighted spaces

Abstract In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp -statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

Construction of New Operators by Composition of Integral-Type Operators and Discrete Operators

In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

Variations on the Choi–Jamiołkowski isomorphism

The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗ -algebras they arise from, which are distinguished by their order of composition. A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism (Choi 1975 Lin. Alg. Appl. 10 285) further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially depends on the dynamical structure in C∗ -algebras. We explain in detail how this dependence gives rise to the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiołkowski (1972 Rep. Math. Phys. 3 275). We then overcome this subtle but pervasive issue in a number of ways: first, we prove a version of Choi’s theorem for Jamiołkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗ -algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.

Generalized Szász-Mirakian Type Operators

In this paper, we are proposing certain modifications of Szász-Mirakian type operators and study their approximation properties. We also give a Voronovskaya type theorem for these operators. Keywords: Linear positive operators, Schurer type operator, Szász-Mirakian operators, Voronovskaya type theorem.

Statistical convergence of integral form of modified Szász–Mirakyan operators: an algorithm and an approach for possible applications

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity.

Bernstein–Kantorovich operators, approximation and shape preserving properties

Inspired by the construction of Bernstein and Kantorovich operators, we introduce a family of positive linear operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{K}_n$$\\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.

Ergodic quantum processes on finite von Neumann algebras

Let (M,τ) be a tracial von Neumann algebra with a separable predual and let (Ω,P) be a probability space. A bounded positive random linear operator on L1(M,τ) is a map γ:Ω×L1(M,τ)→L1(M,τ) so that τ(γω(x)a) is measurable for all x∈L1(M,τ) and a∈M, and x↦γω(x) is bounded, positive, and linear almost surely. Given an ergodic T∈Aut(Ω,P), we study quantum processes of the form γTnω∘γTn−1ω∘⋯∘γTmω for m,n∈Z. Using the Hennion metric introduced in [21], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when γω is the predual of a normal positive linear map on M. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [21].

Asymptotic profiles of basic reproduction number for a degenerate reaction–diffusion host–pathogen model

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN and local basic reproduction number (LBRN) for the model are calculated. Some basic properties of with respect to the general diffusion rates and the asymptotic profiles of with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators and . Our results reveal that is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one.

Extremal structure of cones of positive homogeneous polynomials

It was proved by Anthony Wickstead that the cone of positive linear operators between Banach lattices E and F coincides with the strongly closed convex hull of the set of lattice homomorphisms from E to F if and only if the cone of positive elements on E is the weakly closed convex hull of the union of extremal rays of the cone of positive liner functionals on E. This note aims to show that this result extends to a wider class of operators, namely, orthogonally additive homogeneous polynomials acting between vector lattices.

Estimates related to the iterates of positive linear operators and their multidimensional analogues

Estimates related to the iterates of positive linear operators and their multidimensional analogues

A q‐Erkuş–Srivastava polynomials operator

Chan et al. (Integral Transforms Spec. Funct. 12 (2), 139–148, (2001)) constructed a multivariable extension of the Lagrange polynomials, popularly known as the Chan–Chyan–Srivastava polynomials. Altin and Erkuş (Integral Transforms Spec. Funct. 17, 239–244, (2006)) proposed Lagrange–Hermite polynomials in several variables. Erkuş and Srivastava (Integral Transforms Spec. Funct., 17, 267–273, (2006)) presented an unification (and generalization) of the Chan–Chyan–Srivastava polynomials and the multivariable Lagrange–Hermite polynomials, called as Erkuş–Srivastava polynomials. Duman (Taiwanese J. Math. 12 (2), 539‐543, (2008)) defined a ‐analogue of these generalized multivariable polynomials. Inspired by these studies, we construct a linear positive operator by means of the ‐Erkuş–Srivastava multivariable polynomials and study the Korovkin‐type theorems and the rate of convergence of these operators by using summability techniques of weighted ‐statistical convergence and the power series method. We also define a th‐order Taylor generalization of the multivariable polynomials operator and investigate the approximation of th‐order continuously differentiable Lipschitz class elements. Finally, we define the bivariate case of ‐Erkuş–Srivastava multivariable polynomials operator and study its ‐statistical convergence by using four‐dimensional matrix transformation.

Characterization of Deferred Statistical Convergence of Order $$\alpha $$ for Positive Linear Operators and Application to Generalized Bernstein Polynomials

Characterization of Deferred Statistical Convergence of Order $$\alpha $$ for Positive Linear Operators and Application to Generalized Bernstein Polynomials

Properties of Positive Linear Operators Connected with Squared Fundamental Functions

The aim of this article is to present a general class of positive linear operators of discrete type related to squared fundamental basis functions. If these operators are expressed by a series, we propose to truncate them by a finite sum while still keeping the property of converging towards the identity operator in a weighted space. A relation between the local smoothness of functions and the local approximation is obtained. In the variant in which the operators are described by a finite sum, we establish the limit of their iterates.

Polynomial Weighted Approximation by Szász-Mirakyan Operators of Max-product type

In this paper, we study approximation of Szász-Mirakyan operators of max-product type in polynomial weighted spaces. We reckon the rate of approximation in terms of some exponential weighted spaces for obtain a better rate of approximation than the corresponding positive linear operators.