American Institute of Mathematical Sciences

In this paper, we introduce a bi-variate case of a new kind of \begin{document}$ \lambda $\end{document}-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [31]. 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.

In this paper we deal with bivariate extension of Jain operators. Using elementary method, we show that these opearators are non-increasing in \begin{document}$ n $\end{document} when the attached function is convex. Moreover, we demonstrate that these operators preserve the properties of modulus of continuity. Finally, we present a Voronovskaja type theorem for the sequence of bivariate Jain operators.

Motivated by certain generalizations, in this paper we consider a new analogue of modified Szá sz-Mirakyan-Durrmeyer operators whose construction depends on a continuously differentiable, increasing and unbounded function \begin{document}$ \tau $\end{document} with extra parameters \begin{document}$ \mu $\end{document} and \begin{document}$ \lambda $\end{document}. Depending on the selection of \begin{document}$ \mu $\end{document} and \begin{document}$ \lambda $\end{document}, these operators are more flexible than the modified Szá sz-Mirakyan-Durrmeyer operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem and quantitative estimates for the local approximation.

In this article, we investigate the approximation properties of general cosine-type operators, especially Rogosinski-type operators, in Banach space when there is a cosine operator function. We apply our approach to both trigonometric Rogosinski operators and Shannon sampling operators. Moreover, for some operators, we derived orders of approximation that include numerical estimates of the constants contained in it. We announced a new direction for approximation issues in the Mellin framework.

In this paper we study boundedness properties of certain semi-discrete sampling series in Mellin–Lebesgue spaces. Also we examine some examples which illustrate the theory developed. These results pave the way to the norm-convergence of these operators.

We establish some general Korovkin-type results in cones of set-valued functions and in spaces of vector-valued functions. These results constitute a meaningful extension of the preceding ones.

We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means \begin{document}$ \sigma_n^{\alpha}f $\end{document} of the Fourier series of a multi-dimensional function converge to \begin{document}$ f $\end{document} at each Lebesgue point as \begin{document}$ n\to \infty $\end{document}.

In this paper we put in evidence localization results for the so-called Bernstein max-min operators and a property of translation for the Bernstein max-product operators.