Periodic Korovkin Theorem via $P_{p}^{2}$-Statistical $\mathcal{A}$-Summation Process

In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

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 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.

Korovkin type approximation theorems obtained through generalized statistical convergence

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.

This article is a continuation of our previous works. We mainly investigate a Korovkin type theorem for double sequences of positive linear operators defined in the space of all $2\pi $-periodic and real valued continuous functions on the real two-dimensional space with help of the concept of triangular $A$-statistical convergence, which is a kind of statistical convergence for double real sequences. Also, we analyze the rate of convergence of double operators in this via modulus of continuity.

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.

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity.

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.

Throughout this paper we denote by L p the Banach lattice of p-integrable functions on a σ-finite measure space (X, B, μ), where 1 ≤ p ≤ ∞. We will consider those aspects of the theory of positive linear operators, which are in some way special due to the fact the operators are acting on L p-spaces. For general information about positive operators on Banach lattices we refer to the texts [1]. [20], and [36]. Our focus on L p-spaces does not mean that in special cases some of the results can not be extended to a larger class of Banach lattices of measurable function such as Orlicz spaces or re-arrangement invariant Banach function spaces. However in many cases the results in these extensions are not as precise or as complete as in the case of L p-spaces. We will discuss results related to the boundedness of positive linear operators on L p-spaces. The most important result is the so-called Schur criterion for boundedness. This criterion is the most frequently used tool to show that a concrete positive linear operator is bounded from L p to L q. Then we will show how this result relates to the change of density result of Weis [33]. Next the equality case of Schur’s criterion is shown to be closely related to the question whether a given positive linear operator attains its norm. We discuss in detail the properties of norm attaining operators on L p-spaces and discuss as an example the weighted composition operators on L p-spaces. Then we return to the Schur criterion and show how it can be applied to the factorization theorems of Maurey and Nikisin. Most results mentioned in this paper have appeared before in print, but sometimes only implicitly and scattered over several papers. Also a number of the proofs presented here are new.

The aim of this paper is to present Korovkin theorems for positive linear operators of two variables from $H_{\omega }\left( K\right) $ into $%C_{B}\left( K\right) $ via the power series method. In addition, we give an example that our new approximation result works but its classical case does not work. Furthermore, we obtain the rate of convergence of these operators.

Equi-statistical<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>σ</mml:mi></mml:math>-convergence of positive linear operators

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.

In this paper, we use the power series methods to study Korovkin-type approximation theorems for sequences of operators that are monotone and sublinear. It is important to point out that any positive linear operator is monotone sublinear but the converse is not true. We also calculate the rate of convergence in terms of modulus of continuity. Finally, we provide some examples that satisfy our theorems and give quantitative estimates. We also define Bernstein-Chlodovsky-Kantorovich-Choquet (BCKC) polynomial operators similar to those given before and obtain approximation estimate.

ABSTRACTThe Mihesan operator is the generalization of well‐known positive linear operators like Bernstein operators, Szász–Mirakjan operators, Baskakov operators, and Lupaş operators. In order to expand the theory of positive operators beyond the linear example, nonlinear positive operators are essential. We have introduced the nonlinear Mihesan operators and find the approximation properties of Truncated Mihesan operators with all steps in this note.