A Certain Class of Relatively Equi-Statistical Fuzzy Approximation Theorems

Abstract

The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform statistical convergence as well as relatively deferred Norlund point-wise statistical convergence through the dierence 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 aforesaidnotions. As an application point of view, we establish a fuzzy approximation (Korovkin-type) theorem by using our new notion of relatively deferred Norlund equi-statistical convergence and intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkin-type 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.

A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method

“…Moreover, since ( f m ) is not uniformly statistically convergent to f = 0 over [0, 1] under the deferred power-series technique, and since it is also not simply uniformly convergent, then the classical Korovkin-type theorem does not impartially operate under our recommended operator in (16). Hence, the above notions shows that our Theorem 4 is a non-trivial generalization of some well-established published results (see [21,24,34]).…”

“…Furthermore, Balcerzak et al [33] proposed a stronger result via equi-statistical convergence over the uniform statistical convergence. On the other hand, based upon equi-statistical convergence, different results with various settings have been established by many researchers (see, for example, [20,21,[34][35][36][37][38]). In view of some advanced study in this direction, here, we consider the proposed deferred power-series method in establishing a Korovkin-type theorem, which is based upon the prospective concept of equi-statistically convergence of sequences of functions.…”

“…Influenced by a recently published article by Demirci et al [22], we extract the cognizance of the interested learners concerning the possibilities of establishing some Korvokin-type approximation theorems over the sequence space as well as the probability space. Additionally, in view of the latest result of Paikray et al [34] and Saini and Raj [18], the consciousness of the curious readers is drawn out for future research pertaining to fuzzy approximation theorems.…”