The present article provides a combination of the generalized Sz & aacute;sz-Mirakyan-Durrmeyer operators and the Abel-Ivan type operators. Their compositions provide us some new operators based on associated Laguerre polynomials and the Touchard-type polynomials. We find some convergence and difference estimates of such newly defined operators in certain forms of modulus of continuity.

In the present paper, we study the asymptotic properties of the semi-exponential operator connected with p( x) =x^3 . The main result is a pointwise complete asymptotic expansion valid for locally smooth functions of exponential growth. All coefficients are derived and explicitly given. Furthermore, we characterize classes of functions f for which the semi-exponential operator connected with p( x) =x^3 provides asymptotically a better approximation than the corresponding operator of exponential type. Finally, we present numerical examples which illustrate the better rate of convergence.

In this article, we consider Kantorovich and Durrmeyer-type extensions of the discrete operators defined by İçöz et al. [Filomat 30 (2) (2016), 429–440], which are associated with Miller–Lee polynomials. The primary focus of our study is to obtain the complete asymptotic expansions for these operators. For this purpose, we derive explicit expressions for their moments and central moments in terms of Stirling numbers. Additionally, we examine their convergence behavior, derive quantitative Voronovskaya-type estimates, and establish weighted approximation results. The theoretical results are validated through graphical illustrations and numerical tables.

The motive of this paper is to introduce the generalization of Lupaş-Kantorovich operators connected with Pólya distribution and establish the rate of convergence in terms of modulus of continuity. Furthermore, a Voronovskaja type asymptotic formula for these operators is studied. In the end, few numerical examples with graphical representation are added to depict the effect of convergence of the operators.

The notion of semi-exponential operators broadens the scope of classical exponential-type operators in approximation theory. In the present article, we provide a semi-exponential extension of the exponential operators connected to x^4/3 . The kernel of these integral operators satisfy the partial differential equation, ∂/∂ xϕ _n^β(x,t)=[ n(t-x)/x^4/3-β] ϕ _n^β(x,t) for β > 0 along with normalization condition. We derive the explicit form of the moment generating function and obtain the corresponding moments and central moments. Further, we establish quantitative asymptotic formulae for these operators, along with error estimates. In addition, we provide some numerical examples alongwith graphical representation.

Abstract The present article deals with an approximation operator, which is based on the modified Bessel function of the first kind. Such operators are linear positive operators and preserve only the constant functions among the power test functions. We construct these operators by a partial differential equation (PDE). Also, a more general form based on two parameters is captured from the PDE. We establish some direct approximation results and, in addition, consider the composition of these operators with Szász–Mirakyan operators, which provide us with another new operator, which is based on generalized Laguerre polynomials. We also estimate the approximation process for the new composite operator.

In this study, we introduce and analyze a Kantorovich integral modification of a generalized sequence of positive linear operators, involving an additional parameter β that alters the kernel structure in a non-classical way, with the classical Szász–Kantorovich operator appearing as a special case. We establish the direct and limiting convergence results for the proposed operators and obtain quantitative Voronovskaya-type estimates. Furthermore, approximation properties in Lipschitz-type space are investigated, and a Grüss Voronovskaya-type theorem is derived, highlighting the interaction behavior of the newly introduced Kantorovich operators.

In this article, we provide complete asymptotic expansions of the linear positive operators associated with the generalized Laguerre and Touchard polynomials, respectively. We also provide specific methods to obtain better order of approximation by these operators. Moreover, we establish quantitative difference estimates for these operators with respect to their components through moduli of smoothness. Finally, we investigate their rates of convergence through graphical illustrations.

In this article, we capture Phillips type operators based on adjoint Bernoulli polynomials and their connection to other operators. We derive their characteristic function and provide pointwise convergence and estimate errors using various types of modulus of continuity. Then we establish the theorems based on the difference of operators with their decomposed parts. Additionally, we modify these operators in order to preserve e Ax and e 2Ax and give asymptotic formula and a Korovkin-type result for modified operators.

The present article involves the study of linear positive operators associated with adjoint Apostol-type Frobenius-Euler polynomials. Initially, we introduce the univariate operators and calculate their moments. We establish Korovkin-type theorems using two different sets of test functions. Then we derive a Voronovskaya-type formula for the class of functions having exponential growth. Moreover, we introduce the corresponding bivariate operators and study their pointwise convergence. Additionally, we present quantitative estimates in terms of the first- and second-order moduli of smoothness, as well as the weighted modulus. Finally, we examine the rates of convergence for both operators via graphical examples.

In this article, we introduce a new variant of the Paltanea operator based on modified Hermite polynomials of two variables. We establish several approximation properties for this operator including Voronovskaja-type theorem in weighted space and illustrate its convergence both numerically and graphically. Additionally, we capture a new interesting operator based on a composition method and then establish an asymptotic formula for the composition operator. We also study its convergence in terms of first and second order modulus of continuity and present a theorem based on difference estimates.

In this article, we investigate the convergence behavior of generalized sampling operators of Kantorovich-type. By combining the generalized sampling operators and Kantorovich sampling operators, we obtain the new composition operators and estimate the order of approximation. Then, we establish quantitative estimates for convergence in terms of the first-order modulus of continuity and K$$ K $$-functional. We also estimate the difference of these operators with generalized sampling operators. Moreover, we examine the order of approximation in the weighted space of continuity. Illustrative examples of kernels that meet the necessary assumptions are provided. We also demonstrate the performance of the proposed operators through graphical examples and numerical tables. Finally, we explore their potential applications in digital image processing.

Recently, semi-exponential operators were introduced in literature and within few years it became highly interesting area among researchers. In this direction, we extend the study of semi-exponential Baskakov operators. Here, we capture semi exponential Baskakov type operator by composition method. We establish Voronovskaja’s type results in different spaces, local convergence theorem and present theorems based on difference of operators. Additionally, we derive the convergence for the operators in terms of different type of modulus of continuity. Furthermore, we show the convergence by providing some graphs.

The present paper provides the study on the composition operators due to Szász-Mirakyan with the generalized Szász-Durrmeyer operators. We get an interesting new operator based on Touchard polynomials. Also, the new operator can be further decomposed into two different operators. This study is extension of the recent work, where we have discussed such compositions with reverse order. We find some convergence behavior of these composition operators in sense of point-wise estimates and quantitative estimates in terms of modulus of continuity. We also provide some graphs to have convergence visualization for different n.

The present paper provides the study on the composition operators due to Rathore and Charlier based operators. We consider the composition of the operators based on Charlier polynomials with Rathore operators on both way i.e. M_m^a∘ W_n and W_n ∘ M_m^a . Also, we establish difference of such operators with original ones, error estimation for exponential functions and quantitative asymptotic estimate. We conclude that in limiting case both compositions give same results but the quantitative estimate shows that W_n ∘ M_n^a gives better approximation in comparison to M_n^a∘ W_n . In the end, graphical comparison is indicated.

In the present paper, we study the approximation properties of the compositions of Jain operators and Szász–Mirakjan operators. First, we estimate the moment-generating function and moments of the new operators in terms of the Lambert W function and then we establish some convergence results and their quantitative estimates for the difference of the operators, while emphasizing on the preservation of the modulus of continuity. We further discuss Korovkin-type theorem using a Chebyshev system of exponential functions and Voronovskaja-type asymptotic formulae for these operators. In the last section, we provide a comparative study of the rates of convergence of these operators using graphs and numerical tables.

The integral type exponential operators connected with x^3 were introduced four and half decades ago, but no significant work has been done for many years due to complicated analysis involved to handle such operators. In the recent years, some researchers attracted towards these important operators and they obtained interesting approximation properties. In the present article we discuss operators based on these operators. If we consider the composition of such operators with some discrete operators irrespective of exponential or non-exponential, we capture some new discrete operators. Also, the new operators are based on the modified Bessel’s K functions of second kind. Here we find moments using moment generating function and estimate some convergence results.

In this paper, we consider semi-exponential operators connected to x3. We obtain for them preservation properties, explicit estimates of the rates of convergence, and closed form expressions for their moments. This is done by using probabilistic representations for such operators in terms of expectations involving appropriate random variables.

The present article deals with approximation of a new discretely defined operator based on hypergeometric functions. We study and establish some direct results exponential functions in terms of weighted moduli of continuity.