Simultaneous approximation by certain Baskakov–Durrmeyer–Stancu operators

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.

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.

<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>

On Chlodowsky variant of Szász operators by Brenke type polynomials

The present article is a continuation of the work done by Aral and Erbay [ 1 ]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bogel continuous and Bogel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

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.

<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>

"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."

The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

The goal of this manuscript is to introduce a new sequence of generalized-Baskakov Durrmeyer-Schurer Operators. Further, basic estimates are calculated. In the subsection sequence, rapidity of convergence and order of approximation are studied in terms of first and second-order modulus of continuity. We prove a Korovkin-type approximation theorem and obtain the rate of convergence of these operators. Moreover, local and global approximation properties are discussed in different functional spaces. Lastly, A-statistical approximation results are presented.

In the present paper, we study the convergence of real and modified Post-Widder operators in C, which is known as the extension of approximation of these operators from the real axis in the complex plane. In this direction, we also investigate error estimation in simultaneous approximation and a Voronovskaya-type asymptotic formula.

With the weight of ‐Baskakov basis function, this article considers the approximation properties of the integral modification of exponential type operators. The moments are estimated, and the Voronovskaja‐type result is established in simultaneous approximation. Additionally, the power series method is used for estimating convergence and error estimation. For some functions, the convergence of these operators has also been graphically illustrated.

Although fundamental models have been developed for time extrapolation of creep rupture data, empirical methods are still dominating. One limitation with the empirical methods is that no systematic way of estimating the error in the results has been available. To remove this limitation, formulae for error estimates are developed in the present paper. Error estimates are first established for single creep rupture curves. Then error estimates are formulated for creep data at several test temperatures assessed with Time-Temperature Parameters (TTPs). A new requirement is introduced in the optimization, namely that the first and second derivatives of the rupture curve should be negative, referred to as constrained. The constrained optimization significantly reduces the error in the extrapolated values. The constrained procedure is applied to the austenitic steel Sanicro 25. The procedure gives quite a stable result, and the post-assessment tests developed by ECCC are satisfied automatically for five common TTPs tested.

In this article, we present a generalization of Păltănea operators associating Gould–Hopper polynomials. We obtain approximation properties of our operators by means of the universal Korovkin’s theorem and examine convergence properties by applying modulus of continuity, the second order modulus of smoothness and Peetre’s K-functional and also demonstrated the convergence of these operators graphically for particular functions. Voronovskaja type asymptotic formula is also established. Further, we estimate the rate of point-wise convergence of these operators for functions with derivative of bounded variation. The convergence of these operators are also discussed in weighted spaces of functions on the positive semi-axis and estimate the approximation with the help of weighted modulus of continuity. Lastly, we computed errors in approximating different functions by defined operators and also shown these results graphically. All the graphics and computation of errors are done with the help of MATLAB.

The present study promotes a new algorithm for estimating the water pollution propagation with the primary goal of providing more reliable and high quality estimates to decision makers. To date, the widely used variational method suffers from the large computational burden, which limits its application in practice. Moreover, this method, considering the initial state as a control variable, is very sensitive in specifying initial error, especially for unstable dynamical systems. The Neural Network Filter (NNF), proposed in the present paper, is aimed at overcoming these two drawbacks in the variational method: by its nature, the NNF is sequential (no batch large assimilation window used) and stable even for unstable dynamics, with the gain parameters as control variables. The NNF, developed in the present paper, is a Neural Network Filter (NNF) version of the Singular Evolutive Interpolated Kalman Filter (SEIKF). One of the new versions of this NNF is that it uses structure of the gain of SEIKF0 taken by the SEIKF at the first time moment of correction process. To deal with the uncertainty of the system parameters and of the noise covariance, the proposed Neural Network SEIKF0 named by NNSEIKF0 makes use of the covariance of a reduced rank iterated during assimilation process and of some pertinent gain parameters tuned adaptively to yield the minimum prediction error for the system output. The computational burden in implementation of the NNSEIKF0 is reduced drastically due to applying the optimization tool known as a simultaneous perturbation stochastic approximation (SPSA) algorithm, which requires only two integrations of the numerical model. No iterative loop is required at each assimilation instant as usually happens with the standard gradient descent optimization algorithms. Data assimilation experiment, carried out by the SEIKF0 and NNSEIKF0, is implemented for the Thanh Nhan Lake in Hanoi and the performance comparison between the NNSEIKF0 and SEIKF0 is given to show the high efficiency of the proposed NNSEIKF0.