Generalized Kantorovich type Szász-Mirakjan operators based on q-integers

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.

This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

For a standard Wiener process \(\{ {{W}_{t}}:t\varepsilon [0,1]\}\), denoting \(X(\rho ) = {{\sup }_{{|s - t| \leqslant \rho }}}|{{W}_{s}} - {{W}_{t}}|, g(x) = \sqrt {{2x\ln (\tfrac{1}{x})}}\) and \(h(x) = \sqrt {{2x\ln \ln \tfrac{1}{x}}}\) , we have proved the following two results: \(E {{\sup }_{{\rho \leqslant \delta }}}\tfrac{{X(\rho )}}{{g(\rho )}}\) converges down to 1 at least as fast as \(3.27/\sqrt {{\ln \tfrac{1}{\delta }}}\) and \({{\sup }_{{t \leqslant \delta }}}\tfrac{{{{W}_{t}}}}{{h(t)}}\) converges down to 1 at least as fast as \(3\ln \ln \ln \tfrac{1}{\delta }/\ln \ln \tfrac{1}{\delta }\)

In this article, we consider a bivariate Chlodowsky type Szász–Durrmeyer operators on weighted spaces. We obtain the rate of approximation in connection with the partial and complete modulus of continuity and also for the elements of the Lipschitz type class. Moreover, we examine the degree of convergence with regard to the weighted modulus of continuity and Peetre’s K-functional. Further, we construct the associated GBS type of these operators and estimate the degree of approximation using the mixed modulus of continuity and a class of the Lipschitz of Bögel type continuous functions. Finally, with the help of Maple software, we present the comparisons of the convergence of the bivariate Chlodowsky type Szász–Durrmeyer operators and associated GBS type operators to certain functions with some graphs and error estimation tables.

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.

In this paper we introduce two kinds of Chlodowsky-type q-Bernstein-Schurer-Stancu- Kantorovich operators on the onbounded domain. The Korovkin type statistical approximation property of these operators are investigated. We investigated the rate of convergence for this approximation by means of the first and the second modulus of continuity. The rate of convergence is investigated by using Lipschitz classes of functions and the modulus of continuity of the derivative of the function. Then, we obtain point-wise estimate the Lipchitz type maximal function.

In the present paper, we introduce q-analogue of the Jakimovski-Leviatan operators with the help of q-Appell polynomials. We establish some moments and auxiliary results by using q-derivatives and then prove a basic convergence theorem. Also, the Voronovskaja-type asymptotic formula and some direct results for the above operators are discussed. Moreover, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied.

We introduce Kantorovich variant of Stancu-Lupas? operators and study convergence and qstatistical convergence properties using Korovkin theorem. Rate of convergence is analyzed in terms of modulus of continuity, elements of Lipschitz class and Peetre?s K-functional. Direct theorems are proved and Voronovskaja type theorem is established. Graphical analysis of convergence and error estimations are presented with the help of MATLAB.

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.

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 paper, we introduce the q-analogue of the Jakimovski-Leviatan type modified operators introduced by Atakut with the help of the q-Appell polynomials.We obtain some approximation results via the well-known Korovkin’s theorem for these operators.We also study convergence properties by using the modulus of continuity and the rate of convergence of the operators for functions belonging to the Lipschitz class. Moreover,we study the rate of convergence in terms of modulus of continuity of these operators in aweighted space.

We give quantitative estimates of the rate of convergence of equiuniform approximation processes of integral operators in Banach spaces in terms of the modulus of continuity of functions to be approximated and higher order absolute moments of approximate kernels. Furthermore, applications are presented for various equi-uniform summation processes, interpolation type operators, convolution type operators, and several concrete examples of approximating operators are also provided.

The goal of this research is to construct a generalization of a Kantorovich type of Szász operators involving negative-order Genocchi polynomials. With the aid of Korovkin’s theorem, modulus of continuity, Lipschitz class, and Peetre’s K-functional the approximation properties and convergence rate of these operators are established. To illustrate how operators converge to a certain function, we present some examples.

This article is a continuation of our previous works. We mainly investigate a Korovkin type theorem for double sequences of positive linear operators defined in the space of all $2\pi $-periodic and real valued continuous functions on the real two-dimensional space with help of the concept of triangular $A$-statistical convergence, which is a kind of statistical convergence for double real sequences. Also, we analyze the rate of convergence of double operators in this via modulus of continuity.

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