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Statistical convergence is more extensive than the classical convergence and has recently drawn the recognition of many researchers. The Korovkin-type approximation theorems are usually based on the convergence analysis of sequences of positive linear operators. Gradually, such approximation theorems are extended over more general sequence spaces with several settings via different kinds of statistical summability techniques. In this paper, we introduce presumably a new statistical Riesz-Euler product summability technique to prove a Korovkin-type approximation theorem. Moreover, we demonstrate another result for the rate of statistical convergence under our proposed summability technique. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the notions of deferred Nörlund statistical Riemann integrability and statistical deferred Nörlund Riemann summability for sequence of real-valued functions and to apply them in Korovkin-type new approximations. First, we present an inclusion theorem to understand the connection between these new notions. Then, based on these potential notions we establish new versions of Korovkin-type theorems with three algebraic test functions. Finally, we compute an example, under the consideration of a positive linear operator in association with the Bernstein polynomials to exhibit the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we expose the ideas of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence in the sense of the deferred power-series method. We then propose a relation connecting them, which is followed by several illustrative examples. Moreover, as an application viewpoint, we establish an approximation theorem based upon our proposed method for equi-statistical convergence of sequences of positive linear operators. Finally, we estimate the equi-statistical rates of convergence for the effectiveness of the results presented in our study. [ABSTRACT FROM AUTHOR]

In this article, we introduced the concepts of statistical Riemann integrability, deferred Cesàro statistical Riemann integrability and statistical deferred Cesàro Riemann summability for sequence of real-valued functions. We first established some fundamental theorems connecting these concepts with examples. Then, we developed a criteria for statistical Cauchy–Riemann integrability and a theorem in this connection has been given. Application of the newly introduced Riemann integrability of sequence of functions has been shown with the help of certain new Korovkin-type approximation results, and demonstrated some more interesting results with relevant examples for positive linear operators. We also highlighted some important aspects of our findings in the conclusion section. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and study the concepts of statistical product convergence and statistical product summability via deferred Cesàro and deferred weighted product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two nice and potentially useful concepts. Also, based upon our proposed concepts, we state and prove a set of new Korovkin‐type approximation theorems for a martingale sequence over a Banach space. Moreover, we demonstrate that our approximation theorems effectively extend and improve most (if not all) of the previously existing results (both in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature. We also suggest a direction for future researches on this subject, which are based upon the basic (or q‐) calculus, but not upon the trivial and inconsequential variations involving the so‐called (p, q)‐calculus. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform statistical convergence as well as relatively deferred Nörlund point-wise statistical convergence through the difference operator of fractional order of fuzzy-number-valued sequence of functions, and a type of convergence which lies between aforesaid notions, namely, relatively deferred Nörlund equi-statistical convergence. Also, we investigate the inclusion relations among these aforesaid notions. As an application point of view, we establish a fuzzy approximation (Korovkin-type) theorem by using our new notion of relatively deferred Nörlund equi-statistical convergence and intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkintype theorems which were presented in earlier works. Moreover, we estimate the fuzzy rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function by using the fuzzy modulus of continuity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. 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 real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions. [ABSTRACT FROM AUTHOR]