Direct Estimates for Certain Integral Type Operators

In the present paper, we study approximation properties of a family of linear positive operators and establish direct results, asymptotic formula, rate of convergence, weighted approximation theorem, inverse 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.

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter $$\rho $$ . We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

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 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.

On the Chebyshev-Cramér Asymptotic Expansions

In this paper, following very recent results of Baliarsingh (Alex Eng J 55(2):1811–1816, 2016), we first introduce the concepts of statistically $$\Omega ^{\Delta }$$ -summability and $$\Omega ^{\Delta }$$ -statistical convergence by means of fractional-order difference operator $$\Delta ^{\alpha ,\beta ,\gamma }_{h}$$ . We also present some important inclusion relations between newly proposed methods. Our present investigation deals essentially with various summability techniques and reveals how these methods lead to a number of approximation by positive linear operators. As an application, we prove a Korovkin type approximation theorem and also present an illustrative example using the generating function type Meyer-Konig and Zeller operator. Furthermore, we estimate the rate of convergence of approximating linear operators by means of the modulus of continuity and some Voronovskaja type results are derived. Finally, we present some computational and geometrical interpretations to illustrate some of our approximation results.

We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.

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

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.

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.

In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Norlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Norlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Norlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.

The concept of statistical convergence has attracted the pervasive attention of the current researchers due basically to the fact that it is stronger than the ordinary convergence. Korovkin-type approximation theorem plays a vital role in the convergence of sequences of positive linear operators. Moreover, this type of approximation theorems has been extended through different statistical summability methods over general sequence spaces. The paper investigated statistical $$(C,1)(E,\mu )$$ product summability mean for sequences of fuzzy numbers and proved a fuzzy Korovkin-type approximation theorem. Furthermore, we have established another result for the fuzzy rate of convergence which is uniform in fuzzy Korovkin-type approximation theorem under our proposed summability mean.

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.