This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence.

We propose and analyze an inertial iterative algorithm to approximate a common solution of generalized equilibrium problem, variational inequality problem, and fixed point problem in the framework of a 2-uniformly convex and uniformly smooth real Banach space. Further, we study the convergence analysis of our proposed iterative method. Finally, we give application and a numerical example to illustrate the applicability of the main algorithm.

This manuscript is associated with a study of general Appell polynomials. In this research work, we introduced a new sequence of Szász-Integral type of sequence of operators via general-Appell polynomials to discuss approximation properties for Lebesgue integrable functions $ (L^p[0, \infty)) $. In addition, estimates are studied in view of test functions and central moments. Next, the convergence rate is discussed using the Korovkin theorem and a Voronovskaja-type theorem. Moreover, direct approximation results via modulus of continuity of first and second order, Peetre's K-functional, Lipschitz type space, and the $ r^{th} $ order Lipschitz type maximal functions are investigated. In the following section, we present weighted approximation results, and statistical approximation theorems are discussed.

This research work focuses on $\lambda$-Szász–Mirakjan operators coupling generalized beta function. The kernel functions used in $\lambda$-Szász operators often possess even or odd symmetry. This symmetry influences the behavior of the operator in terms of approximation and convergence properties. The convergence properties, such as uniform convergence and pointwise convergence, are studied in view of the Korovkin theorem, the modulus of continuity, and Peetre’s K-functional of these sequences of positive linear operators in depth. Further, we extend our research work for the bivariate case of these sequences of operators. Their uniform rate of approximation and order of approximation are investigated in Lebesgue measurable spaces of function. The graphical representation and numerical error analysis in terms of the convergence behavior of these operators are studied.

The goal of this manuscript is to introduce a new sequence as (σ, μ)-Stancu-Schurer operators. Further, some estimates are calculated as test functions and central moments. In the next section, rapidity of convergence and order of approximation are studied in terms of first and second order modulus of continuity. Voronovskaja-type approximation result is investigated to approximate the class of functions which has first and second order continuous derivatives. Moreover, local approximation properties are studied in several functional spaces. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically.