Approximation properties of generalized Baskakov operators

Introduce the notation: $$\mathbb{Z}$$ is the set of integers, $$\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$$ , $$g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$$ , where $$g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $$ . Under certain conditions on the functions g 1, g 2, we prove that the system of functions $$g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$$ , where $$\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $$ are arbitrary infinitesimal sequences, is complete in the space C $$\mathbb{R}^2 $$ of uniformly continuous bounded functions f equipped with the norm $$||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$$ . Starting with the functions g k,m , it is possible to construct a method for uniform approximating in $$\mathbb{R}^2 $$ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.

In this paper, we introduce the Bezier variant of two new families of generalized Bernstein type operators. We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity. By means of construction of suitable functions and the method of Bojanic and Cheng, we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

<p style='text-indent:20px;'>In this paper, we introduce a bi-variate case of a new kind of <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [<xref ref-type="bibr" rid="b31">31</xref>]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.</p>

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

On Chlodowsky variant of Szász operators by Brenke type polynomials

"In this paper we remark that alfa-Bernstein operators, introduced by X. Y. Chen et al., are combinations of two known operators (Stancu and Bernstein operators) and we establish the preservation of global smoothness properties by these linear operators, the global smoothness being expressed by a Lipschitz condition with a certain second order modulus of continuity."

We obtain estimates for the transformation of the second order modulus of continuity by positive linear operators which satisfy certain conditions, which are common to a large class of approximation operators.

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

In this study we construct the GBS (Generalized Boolean Sum) operators associated with combination of Chlodowsky and modified Sz?sz operators and estimate the degree of approximation for these operators in terms of the mixed (B?gel) modulus of smoothness. Furthermore, we improve the measure of smoothness by the mixed K-functional.

We introduce bivariate case of the Szasz-type operators based on multiple Appell polynomials introduced by Varma (Stud. Univ. Babes -Bolyai Math. 58, 361–369 (2013)). We establish a uniform convergence theorem and determine the degree of approximation in terms of the partial moduli of continuity of the approximated function. We estimate the error in simultaneous approximation of the function by the bivariate operators by using finite differences. We investigate the degree of approximation of the bivariate operators by means of the Peetre’s K-functional. The rate of convergence of these operators is determined for twice continuously differentiable functions by Voronovskaja-type asymptotic theorem. The weighted approximation properties are derived for unbounded functions with a polynomial growth. Lastly, we introduce the associated generalized boolean sum (GBS) of the bivariate operators to study the approximation of Bogel-continuous and Bogel-differentiable functions and establish the approximation degree with the aid of the Lipschitz class of Bogel-continuous functions and the mixed modulus of smoothness.

In this paper we will obtain some Bernstein-Kantorovich operators modified in Stancu sense which preserve exponential function eμx, where μ > 0. Concerning these operators we prove they verify Korovkin’s theorem conditions and also that they approximate functions from a weighted Lp space. Moreover, we will obtain a Voronovskaya theorem and some quantitative estimates of approximation using the first order modulus of continuity. Also, we will prove some estimates concerning the approximation of functions from a weighted Lp space using Peetre’s K-functional. Finally, we will obtain an estimate which involves the first order modulus of continuity and the second order modulus of smoothness by using the equivalence relation between these moduli and the corresponding K-functionals.

Neural networks are widely used in many applications including astronomical physics, image processing, recognition, robotics, and automated target tracking, etc. Their ability to approximate arbitrary continuous functions is the main reason for this popularity. The authors of this paper show by constructive methods that for any continuous function f on [a, b], the function can be approximated by a feedforward network with one hidden layer of sigmoidal neurons, and the first order modulus of continuity is used to estimate the approximation error. In particular, when the activation functions of hidden layer are satlin function, we give the constructive proofs for the neural network to reach the second order modulus of continuity result. Using the univariate constructive technique of the neural network, we present a multivariate interpolation neural network. Some numerical results are given to demonstrate the theoretical results.

<p style='text-indent:20px;'>The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [<xref ref-type="bibr" rid="b20">20</xref>]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the <inline-formula><tex-math id="M1">\begin{document}$ (2k+2)th $\end{document}</tex-math></inline-formula> order modulus of continuity using Steklov mean.</p>

In the present paper, we establish some direct results in simultaneous approximation for Baskakov–Durrmeyer–Stancu (abbr. BDS) operators Dn(α,β)(f,x). We establish point-wise convergence, Voronovskaja type asymptotic formula and an error estimate in terms of second order modulus of continuity of the function.

Many papers introduced for the approximation of continuous functions on a closed intervals using algebraic polynomials. Here we shall introduce a sequence of trigonometric operators on the Lebesgue quasi normed spaces, Then we using it to approximate functions in Lebesgue quasi normed spaces, in terms of modulus of smoothness of degree k.