Generalized Bernstein–Durrmeyer operators of blending type

In this note, we study approximation properties of a family of linear positive operators and establish asymptotic formula, rate of convergence, local approximation theorem, global approximation theorem, weighted approximation theorem, and better approximation for this family of linear positive operators.

Stancu generalized Baskakov operators using inverse Polya–Eggenberger distribution for a real valued bounded function on $$[0,\infty )$$ and a non-negative real number $$\alpha $$ (which may depend on n). Dhamija and Deo (Appl Math Comput 286:15–22, 2016) introduced a Durrmeyer type modification of these generalized operators and studied the uniform convergence, the rate of convergence by means of the moduli of continuity and the Peetre’s K-functional. The purpose of the present paper is to continue to study the approximation properties of these Durrmeyer type operators. Local and global direct approximation theorems and a Voronovskaya type asymptotic theorem are established. The quantitative Voronovskaya and Gruss Voronovskaya type theorems are investigated by calculating the indispensible sixth order moment. The weighted approximation properties and the approximation of functions with derivatives of bounded variation are also studied.

Convergence of scaled vector fields and local approximation theorem on Carnot-Carathéodory spaces and applications

In this paper, a generalized sequence of Baskakov operators connected to the Riemann–Liouville fractional integral is introduced. These sequences of operators deal with smooth approximation behavior in a wider class, i.e., a class of measurable functions in the Lebesgue sense. Further, estimates in terms of test functions and central moments are established. Using the weighted Korovkin-type theorem, uniform convergence in weighted spaces is studied. Additionally, uses of the classical modulus of continuity and Peetre’s K-functional to establish local and global direct approximation theorems are introduced. Lastly, the statistical convergence properties of the suggested operators are investigated. In the last section, theoretical results are supported by numerical examples with graphical and numerical illustrations.

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.

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 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.

Exponential polynomials satisfying a homogeneous convolution type equation are called its elementary solutions. The traditional solution of a homogeneous equation of the convolution type involves the proof of an approximation theorem that asserts the density of elementary solutions in the set of all solutions of the equation. This article considers homogeneous equations of the π-convolution type in the space of analytic functions on a convex domain, which generalize the well-known homogeneous equations of the convolution type. Equations of a particular form have been considered many times before. Such equations include homogeneous convolution equations, homogeneous q-sided convolution equations, homogeneous q-sided convolution type equations, and homogeneous-convolution equations. In the article the properties of operators of π-convolution type are investigated and an approximation theorem for a homogeneous equation of π-convolution type in an arbitrary convex domain of the complex plane is proved.

This paper is primarily concerned with approximation at 0 of an (a, k)-regularized resolvent family R(·) for Volterra integral equation. We shall consider convergence rates of some kind of local means Qm(t), t ≥ 0,m ≥ 0, of R(t)/k(t). Some approximation theorems and local ergodic theorems with rates will be deduced from general approximation theorems for regularized approximation processes.

Convergence of a family of neural network operators of the Kantorovich type

In this paper, we introduce a new kind of modified Bernstein–Schurer operators based on the concept of (p, q)-integers. We investigate statistical approximation properties, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Next, we construct the bivariate operators and get some convergence properties. Finally, we give some graphs to illustrate the convergence properties of operators to some functions.

ABSTRACTThis work introduces ‐Bézier–Bernstein‐integral type operators, along with local approximation results, a direct approximation theorem that leverages the Ditzian–Totik modulus of smoothness, and a quantitative Voronovskaja‐type theorem using the Ditzian–Totik modulus of continuity. Additionally, we derive the convergence rate for differential functions with bounded variation in their derivatives. Ultimately, we demonstrate numerical examples that validate the theoretical results and showcase the effectiveness of the defined operators.

In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators $$L_n^{j,i}$$ L n j , i , $$i,j\in {\mathbb {N}}_0$$ i , j ∈ N 0 . For $$i=j=1$$ i = j = 1 , these operators reduce to the genuine Bernstein–Durrmeyer operators, while for $$i=j=0$$ i = j = 0 they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for $$L_n^{j,i}$$ L n j , i under suitable smoothness conditions on the approximated function.

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.

We use the new variant of Szász–Mirakjan operators to construct a generalized version of Szász‐beta type operators and obtain auxiliary lemmas. We present the weighted approximation theorems and, by using Peetre’s K ‐function, the local approximation results of these operators are studied. We also analyze the rate of convergence by utilizing Lipschitz‐type maximal functions and the ordinary modulus of continuity. Moreover, using aforesaid operators, we finally examine the Voronovskaya‐type theorem.