Statistical deferred Cesàro summability and its applications to approximation theorems

Abstract

Statistical (C, 1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C, 1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Cesàro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e −x and e −2x defined on a Banach space C[0, ∞) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Cesàro summability method. Some interesting examples are also discussed here in support of our definitions and results.

Strong Riesz summability of Fourier series

“…Also, the statistical convergene and statistical summability are more general than the ordinary convergene and ordinary summability. For recent works in this direction, see [1], [5], [6], [8], [9], [10], [11], [15], [16], [17], [18], [19] and [20]. Dealing with strong summability of Nörlund mean Mittal [13] has proved a theorem as follows.…”

Generalized Nörlund and Nörlund-type Means of Sequences of Fuzzy Numbers

“…Also, Das et al [2] used statistical (C, 1)(E, µ) product summability mean for sequences of fuzzy numbers to prove a fuzzy Korovkin-type approximation theorem. For more studies in this direction one may refer to [3], [4], [5], [7], [8], [14], [15] and [16]. Motivated essentially by the above mentioned works, we investigate here the characterization of generalized Nörlund and Nörlund-type (Riesz) means of sequences of fuzzy real numbers.…”

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

“…Furthermore, Srivastava et al [23,24], studied the notion of the deferred weighted as well as deferred Nörlund statistical convergence and used these notions to prove certain Korovkin-type approximation theorem with some new settings. Later on, some fundamental concept of the deferred Cesàro statistical convergence as well as the statistical deferred Cesàro summability, together with the associated approximation theorems was introduced by Jena et al [25]. In 2019, Kandemir [26] studied the I-deferred statistical convergence in topological groups.…”