Local Approximation Theorem Research Articles

Bivariate $ \lambda $-Bernstein operators on triangular domain

<abstract><p>This paper introduced a novel class of bivariate $ \lambda $-Bernstein operators defined on triangular domain, denoted as $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters $ \lambda_1 $ and $ \lambda_2 $. A Korovkin-type approximation theorem for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ was established, with the convergence rate being characterized by both the complete and partial moduli of continuity. Additionally, a local approximation theorem and a Voronovskaja-type asymptotic formula were derived for $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $. Finally, the convergence of $ B_{m}^{\lambda_1, \lambda_2}(f; x, y) $ to $ f(x, y) $ was illustrated through graphical representations and numerical examples, highlighting instances where they surpass the performance of standard bivariate Bernstein operators defined on triangular domain, $ B_{m}(f; x, y) $.</p></abstract>

Approximation by the modified $ \lambda $-Bernstein-polynomial in terms of basis function

<abstract><p>In this article by means of shifted knots properties, we introduce a new type of coupled Bernstein operators for Bézier basis functions. First, we construct the operators based on shifted knots properties of Bézier basis functions then investigate the Korovkin's theorem, establish a local approximation theorem, and provide a convergence theorem for Lipschitz continuous functions and Peetre's $ K $-functional. In addition, we also obtain an asymptotic formula of the type Voronovskaja.</p></abstract>

A new type of Szász–Mirakjan operators based on q-integers

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Genuine q-Stancu-Bernstein–Durrmeyer Operators

In the present paper, we introduce the genuine q-Stancu-Bernstein–Durrmeyer operators Znq,α(f;x). We calculate the moments of these operators, Znq,α(tj;x) for j=0,1,2, which follows a symmetric pattern. We also calculate the second order central moment Znq,α((t−x)2;x). We give a Korovkin-type theorem; we estimate the rate of convergence for continuous functions. Furthermore, we prove a local approximation theorem in terms of second modulus of continuity; we obtain a local direct estimate for the genuine q-Stancu-Bernstein–Durrmeyer operators in terms of Lipschitz-type maximal function of order β and we prove a direct global approximation theorem by using the Ditzian-Totik modulus of second order.

Higher order Kantorovich-type Szász–Mirakjan operators

In this paper, we define new higher order Kantorovich-type Szász–Mirakjan operators, we give some approximation properties of these operators in terms of various moduli of continuity. We prove a local approximation theorem, a Korovkin-type theorem, and a Voronovskaja-type theorem. We also prove weighted approximation theorems for these new operators.

Approximation Theorem for New Modification ofq-Bernstein Operators on (0,1)

In this work, we extend the works of F. Usta and construct new modifiedq-Bernstein operators using the second central moment of theq-Bernstein operators defined by G. M. Phillips. The moments and central moment computation formulas and their quantitative properties are discussed. Also, the Korovkin-type approximation theorem of these operators and the Voronovskaja-type asymptotic formula are investigated. Then, two local approximation theorems using Peetre’sK-functional and Steklov mean and in terms of modulus of smoothness are obtained. Finally, the rate of convergence by means of modulus of continuity and three different Lipschitz classes for these operators are studied, and some graphs and numerical examples are shown by using Matlab algorithms.

$$\alpha $$-Bernstein–Kantorovich operators

In this paper we proposed the Kantorovich form of $$\alpha $$-Bernstein operators introduced by Chen et al. (Math Anal Appl 450:244–261, 2017). We give some auxiliary properties and study the direct local approximation theorem, Voronovskaya type asymptotic and function of bounded variation for $$\alpha $$-Bernstein–Kantorovich operators.

Generalized Szász-Kantorovich Type Operators

In this note, we present Kantorovich modification of the operators introduced by V. Mihe s an [ Creative Math. Inf. 17 (2008), 466 – 472]. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. Furthermore, we show the rate of convergence of these operators to certain functions with the help of the illustrations using Maple algorithms.

Generalized Bernstein–Durrmeyer operators of blending type

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter $$\rho $$ . We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

(p,q)-gamma operators which preserve x^{2}

In this paper, we introduce (p,q)-gamma operators which preserve x^{2}, we estimate the moments of these operators, and establish direct and local approximation theorems of these operators. Then two approximation theorems about Lipschitz functions are obtained. The estimates on the rate of convergence and some weighted approximation theorems of the operators are also obtained. Furthermore, the Voronovskaja-type asymptotic formula is also presented.

Longitudinal Spin Fluctuations and Magnetic Ordering Temperature in Metals: First-Principle Modeling and Phase Space Integration Measure

In magnetically ordered metals the magnitude of the local atomic moment become temperature dependent. To deal with this problem on the ab-initio level one need to employ a specific methodology for calculation of the electronic structure that takes into the account the magnetic disorder effects. In addition one needs to setup a special statistical models allowing simultaneously for ab-initio mapping and for the variation of the local spin magnitude. To this end here we discuss and employ methodology that is based on the Disordered Local Moment (DLM) formalism, spin-constraint Local Spin Density Approximation (LSDA) and Lichtenstein theorem for calculation of the inter-site exchange interactions. An extended classical Heisenberg Hamiltonian used for mapping allows for the variation of the lattice site spin magnitude. We consider here three representative canonical transition metals ferromagnets hcp Gd, bcc Fe and fcc Ni with quite a different character of the magnetic moment localization and illustrate the relative importance of the longitudinal spin fluctuations and the magnetic disorder induced electronic structure reconstruction. We use recently introduced linear measure [1] for integration over the longitudinal spin component in the classical configurational spin space.

Direct Estimates for Certain Integral Type Operators

In this note, we study approximation properties of a family of linear positive operators and establish asymptotic formula, rate of convergence, local approximation theorem, global approximation theorem, weighted approximation theorem, and better approximation for this family of linear positive operators.

The Local Approximation Theorem in Various Coordinate Systems

We find some sufficient conditions on the local coordinate system of a Carnot–Caratheodory space of low smoothness that ensure Gromov-type estimates on the divergence of local (quasi)metrics. We also obtain these estimates for the canonical coordinate system of the second kind and various mixed coordinate systems.

On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

We construct a new family of univariate Chlodowsky type Bernstein-Stancu-Schurer operators and bivariate tensor product form. We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity. We also give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions and show that the new ones have a better approximation to functions f for one dimension.

Jain–Durrmeyer Operators Involving Inverse Pólya–Eggenberger Distribution

Stancu generalized Baskakov operators using inverse Polya–Eggenberger distribution for a real valued bounded function on $$[0,\infty )$$ and a non-negative real number $$\alpha $$ (which may depend on n). Dhamija and Deo (Appl Math Comput 286:15–22, 2016) introduced a Durrmeyer type modification of these generalized operators and studied the uniform convergence, the rate of convergence by means of the moduli of continuity and the Peetre’s K-functional. The purpose of the present paper is to continue to study the approximation properties of these Durrmeyer type operators. Local and global direct approximation theorems and a Voronovskaya type asymptotic theorem are established. The quantitative Voronovskaya and Gruss Voronovskaya type theorems are investigated by calculating the indispensible sixth order moment. The weighted approximation properties and the approximation of functions with derivatives of bounded variation are also studied.

Approximation properties of \u03bb-Bernstein operators

In this paper, we introduce a new type λ-Bernstein operators with parameter lambdain[-1,1], we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of B_{n,lambda }(f;x) to f(x), and we see that in some cases the errors are smaller than B_{n}(f) to f.

Approximation Properties of Generalized Szász-Type Operators

In the present paper, we study some approximation properties of the generalized Szasz type operators introduced by V. Mihesan (Creat. Math. Inf. 17:466–472, 2008). We present a quantitative Voronovskaya-type theorem, local approximation theorem by means of second-order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.

A new modification of Durrmeyer type mixed hybrid operators

In 2008 V. Mihes¸an constructed a general class of linear positive operators generalizing the Szasz operators. In ´ this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.

Quantitative-Voronovskaya and Gr\xfcss-Voronovskaya type theorems for Sz\xe1sz-Durrmeyer type operators blended with multiple Appell polynomials

In this paper, we establish a link between the Szász-Durrmeyer type operators and multiple Appell polynomials. We study a quantitative-Voronovskaya type theorem in terms of weighted modulus of smoothness using sixth order central moment and Grüss-Voronovskaya type theorem. We also establish a local approximation theorem by means of the Steklov means in terms of the first and the second order modulus of continuity and Voronovskaya type asymtotic theorem. Further, we discuss the degree of approximation by means of the weighted spaces. Lastly, we find the rate of approximation of functions having a derivative of bounded variation.

A genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions

In this paper, we construct a genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions. We establish some moment estimates and the direct results which include global approximation theorem in terms of classical modulus of continuity, local approximation theorem in terms of the second order Ditizian-Totik modulus of smoothness, Voronovskaya-type asymptotic theorem and a quantitative estimate of the same type. Lastly, we study the approximation of functions having a derivative of bounded variation.