Statistical αβ-summability and Korovkin type approximation theorem

Modularly weighted four dimensional matrix summability with application to Korovkin type approximation theorem

In this paper, we investigated the notion of generalized weighted λ-statistical convergence connecting $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability of sequences of fuzzy numbers. Some inclusion relations and other related results for the new summability methods are also given. Furthermore, we give a Korovkin-type approximation theorem for fuzzy positive linear operators using the notion of generalized weighted $$ \left( {\bar{N}_{\lambda } , p_{n} } \right)^{F} $$ -statistical summability.

In the present paper, Durrmeyer typeλ-Bernstein operators via (p,q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in someλcases than Durrmeyer type (p,q)-Bernstein operators.

The concept of f-lacunary statistical convergence which is, in fact, a generalization of lacunary statistical convergence, has been introduced recently by Bhardwaj and Dhawan (Abstr. Appl. Anal. 2016:9365037, 2016). The main object of this paper is to prove Korovkin type approximation theorems using the notion of f-lacunary statistical convergence. A relationship between the newly established Korovkin type approximation theorems via f-lacunary statistical convergence, the classical Korovkin theorems and their lacunary statistical analogs has been studied. A new concept of f-lacunary statistical convergence of degree β (0 < beta< 1) has also been introduced, and as an application a corresponding Korovkin type theorem is established.

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.

Korovkin type approximation theorems are useful tools to check whether a given sequence ( L n ) n ≥ 1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and x 2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability (C,1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.MSC:41A10, 41A25, 41A36, 40A30, 40G15.

Korovkin type approximation theorems obtained through generalized statistical convergence

Following the notion of $A^\mathcal{I}$-summability method for real sequences \cite{espdsd2} we establish a Korovkin type approximation theorem for positive linear operators on $UC_{*}[0,\infty)$, the Banach space of all real valued uniform continuous functions on $ [0,\infty)$ with the property that $\displaystyle{\lim_{x\rightarrow \infty}f(x)}$ exists finitely for any $f\in UC_{*}[0,\infty)$. In the last section, we extend the Korovkin type approximation theorem for positive linear operators on $UC_{*}\left([0,\infty)\times[0,\infty)\right)$. We then construct an example which shows that our new result is stronger than its classical version.

In this study, a new kind of modified λ-Bernstein-Stancu operators is constructed. Compared with the original λ-Bézier basis function, the newly operator basis function is more concise in form and has certain symmetry beauty. The moments and central moments are computed. A Korovkin-type approximation theorem is presented, and the degree of convergence is estimated with respect to the modulus of continuity, Peetre’s K-functional, and functions of the Lipschitz-type class. Moreover, the Voronovskaja type approximation theorem is examined. Finally, some numerical examples and graphics to show convergence are presented.

Generalized weighted statistical convergence and application

We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence.

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Norlund summability mean. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by the generalized Meyer-Konig and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

In this work, we obtain a Korovkin-type approximation theorem for double sequences of real-valued functions by using the power series method in two-dimensional weighted spaces. We also study the rate of convergence by using the weighted modulus of continuity, and in the last section, we present an application that satisfies our new Korovkin-type approximation theorem but does not satisfy classical one.

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings.

Statistical summability and approximation by de la Vallée-Poussin mean