Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

Abstract

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials in computational disciplines, we propose a new generalization of Bernstein–Kantorovich operators involving shape parameters λ,α and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program.

A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator

“…As shown in Table 1 , compared with the classical BK operator [ 18 ], -BK [ 24 ] ( ), -BK [ 19 , 30 ] ( ), and -BK [ 31 ] ( ) make a comparison, and the error of the BKS operator proposed in this paper is smaller. Under the condition that of the model in this paper is satisfied, the error is the smallest.…”

“…They are mainly used to optimize the process of updating controller parameters to make it approach quickly and ensure accuracy [ 16 ]. Moreover, with the development of the BK operator, some improved BK operators, such as λ -BK [ 17 , 18 ], -BK [ 19 ], -BK [ 20 ], and other methods, have also been developed [ 21 , 22 , 23 , 24 ]. In this paper, a new BKS operator, which is introduced in [ 25 ], is implemented in the manipulator controller to optimize the parameter update process of the disturbance observer and improve the trajectory tracking accuracy of the robotic manipulator controller.…”

Modüler Uzaylar Üzerinde P-İstatistiksel A-Toplam Süreci Aracılığıyla Yaklaşım

“…There is a wide variety of Korovkin type results in the approximation (see e.g. [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]). Korovkin-type approximation first obtained by Bardaro and Mantellini [21] in the abstract setting of the modular spaces, a class of function spaces which includes weighted spaces, certain interpolation spaces, Orlicz and Musielak-Orlicz spaces, L p spaces.…”

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

“…Mohiuddine et al [6], Acu et al [7], İçöz and Çekim [8,9], and Kajla and Micláus [10,11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in diferent functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] in the same feld. In the recent past, for g ∈ [0, 1], m ∈ N and α ∈ [−1, 1], Chen et al [31] constructed a sequence of new linear positive operators as…”