Approximation by a new sequence of operators involving Apostol-Genocchi 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.

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.

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.

<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>

<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.

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.

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

A class of robust smoother and interpolator algorithms is introduced. The motivation for these smoothers and interpolators is a theorem concerning approximate conditional-mean smoothers for vector Markov processes in additive non-Gaussian noise. This theorem is the smoothing analog of Masreliez’s approximate non-Gaussian filter theorem (IEEE-Auto. Control, AC-20, 1975). The theorem presented here relies on the assumption that a certain conditional density is Gaussian, just as does Masreliez’s result. This assumption will rarely, if ever, be satisfied exactly. Thus a continuity theorem is also presented which lends support to the intuitive notion that the conditional density in question will be nearly Gaussian in a strong sense when the additive noise is nearly Gaussian in a comparatively weak sense. Approaches for implementing the robust smoothers and interpolators is discussed and an application to a real data set is presented.

Modulus of continuity and Lipschitz approximation

In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

The space-continuous–discontinuous Galerkin method

Weighted statistical convergence based on generalized difference operator involving (p,q)-gamma function and its applications to approximation theorems