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

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## Abstract

**:**

## 1. Introduction, Preliminaries and Motivation

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Example**

**1.**

- (a)
- If, for each $\u03f5>0$ and for every $x\in I,$$$\begin{array}{c}\hfill lim\frac{{\mathcal{K}}_{m}(x,\u03f5)}{m}=0\phantom{\rule{1.em}{0ex}}(m\to \infty ),\end{array}$$$$\begin{array}{c}\hfill {\mathcal{K}}_{m}(x,\u03f5):=\left|\right\{m\in \left({a}_{n}{b}_{n}\right]\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}|{f}_{m}\left(x\right)-f\left(x\right)|\geqq \u03f5\}|,\end{array}$$$$\begin{array}{c}\hfill {f}_{m}\to f\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\mathrm{stat}-\mathrm{pointwise}\right).\end{array}$$
- (b)
- If, for each $\u03f5>0$,$$\begin{array}{c}\hfill \underset{m\to \infty}{lim}\frac{{\mathcal{K}}_{m}(x,\u03f5)}{m}=0\mathrm{uniformly}\mathrm{with}\mathrm{regards}\mathrm{to}x\in I,\end{array}$$$$\begin{array}{c}\hfill {f}_{m}\u27f9f(\mathrm{equi}-\mathrm{stat}).\end{array}$$
- (c)
- If, for each $\u03f5>0,$$$\begin{array}{c}\hfill \underset{m\to \infty}{lim}\frac{{\mathcal{D}}_{m}\left(\u03f5\right)}{m}=0,\end{array}$$$$\begin{array}{c}\hfill {D}_{m}\left(\u03f5\right)=\left|\right\{m\in \left({a}_{n}{b}_{n}\right]\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\Vert {f}_{m}{-f\Vert}_{C\left(I\right)}\geqq \u03f5\left\}\right|,\end{array}$$$$\begin{array}{c}\hfill {f}_{m}\rightrightarrows f(\mathrm{stat}-\mathrm{unifomly}).\end{array}$$

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Lemma**

**1.**

**Example**

**2.**

**Example**

**3.**

## 2. A Korovkin-Type Approximation Theorem

**Theorem**

**1**

**.**Let $\left({\mathfrak{L}}_{m}\right)$ be the sequences of (positive) linear operators on $C\left(I\right)$, and then, for each $f\in C\left(I\right)$,

**Theorem**

**2**

**.**Let $\left({\mathfrak{L}}_{m}\right)$ be the sequences of positive linear operators on $C\left(I\right)$, and then, for each $f\in C\left(I\right)$,

**Theorem**

**3**

**.**Let $\left({\mathfrak{L}}_{m}\right)$ be a sequence of (positive) linear operators on $C\left(I\right)$, and then, for all $f\in C\left(I\right),$

**Theorem**

**4.**

**Proof.**

## 3. Geometrical View of Theorem 4

**Example**

**4.**

## 4. Rate of DP-Equi-Statistical Convergence

**Definition**

**8.**

**Lemma**

**2.**

- (i)
- $[{f}_{m}\left(x\right)+{g}_{m}\left(x\right)]-[f\left(x\right)+g\left(x\right)]=o\left({d}_{m}\right)$$(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})$ on I
- (ii)
- $[{f}_{m}\left(x\right)-f\left(x\right)][{g}_{m}\left(x\right)-g\left(x\right)]=o\left({t}_{m}{c}_{m}\right)$$(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})\mathrm{on}I$
- (iii)
- $\lambda [{f}_{m}\left(x\right)-f\left(x\right)]=o\left({t}_{m}\right)$$(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})\mathrm{on}I$ for any scalar $\lambda $
- (iv)
- $\sqrt{|{f}_{m}\left(x\right)-f\left(x\right)|}=o\left({t}_{m}\right)$$(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})\mathrm{on}I$,

**Proof.**

**Theorem**

**5.**

- (i)
- ${\mathfrak{L}}_{m}(1,x)-1=o\left({t}_{m}\right)\phantom{\rule{1.em}{0ex}}(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})\mathrm{on}I$
- (ii)
- $\omega (f,{\mu}_{m})=o\left({c}_{m}\right)\phantom{\rule{1.em}{0ex}}(\mathrm{equi}-{\mathrm{stat}}_{\mathrm{DP}})\mathrm{on}I,$

**Proof.**

## 5. Concluding Remarks and Observations

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Zygmund, A. Trigonometric Series; Cambridge University Press: Cambridge, UK, 1959. [Google Scholar]
- Fast, H. Sur la convergence statistique. Colloq. Math.
**1951**, 2, 241–244. [Google Scholar] [CrossRef] - Steinhaus, H. Sur la convergence ordinaire et la convergence asymtotique. Colloq. Math.
**1951**, 2, 73–74. [Google Scholar] - Jena, B.B.; Paikray, S.K. Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems. Univ. Sci.
**2020**, 25, 409–433. [Google Scholar] [CrossRef] - Jena, B.B.; Paikray, S.K.; Dutta, H. A new approach to Korovkin-type approximation via deferred Cesàro statistical measurable convergence. Chaos Solitons Fractals
**2021**, 148, 111016. [Google Scholar] [CrossRef] - Agnew, R.P. On deferred Cesàro means. Ann. Math.
**1932**, 33, 413–421. [Google Scholar] [CrossRef] - Jena, B.B.; Paikray, S.K.; Misra, U.K. Statistical deferred Cesàro summability and its applications to approximation theorems. Filomat
**2018**, 32, 2307–2319. [Google Scholar] [CrossRef] - Boos, J. Classical and Modern Methods in Summability; Clarendon (Oxford University) Press: Oxford/London, UK; New York, NY, USA, 2000. [Google Scholar]
- Šalát, T. On statistically convergent sequences of real numbers. Math. Slovaca
**1980**, 30, 139–150. [Google Scholar] - Maddox, I.J. Statistical convergence in a locally convex space. Math. Proc. Camb. Phil. Soc.
**1988**, 104, 141–145. [Google Scholar] [CrossRef] - Fridy, J.A.; Orhan, C. Lacunary statistical summability. J. Math. Anal. Appl.
**1993**, 173, 497–504. [Google Scholar] [CrossRef] - Móricz, F. Tauberian conditions under which statistical convergence follows from statistical summability (C, 1). J. Math. Anal. Appl.
**2002**, 275, 277–287. [Google Scholar] [CrossRef] - Mohiuddine, S.A.; Alotaibi, A.; Mursaleen, M. Statistical summability (C, 1) and a Korovkin-type approximation theorem. J. Inequal. Appl.
**2012**, 2012, 172. [Google Scholar] [CrossRef] - Karakaya, V.; Chishti, T.A. Weighted statistical convergence. Iran. J. Sci. Technol. Trans. A Sci.
**2009**, 33, 219–223. [Google Scholar] - Mursaleen, M.; Karakaya, V.; Ertürk, M.; Gxuxrsoy, F. Weighted statistical convergence and its application to Korovkin-type approximation theorem. Appl. Math. Comput.
**2012**, 218, 9132–9137. [Google Scholar] [CrossRef] - Baliarsingh, P. On statistical deferred A-convergence of uncertain sequences. Int. Uncertain. Fuzziness-Knowl.-Based Syst.
**2021**, 29, 499–515. [Google Scholar] [CrossRef] - Saini, K.; Raj, K.; Mursaleen, M. Deferred Cesàro and deferred Euler equi-statistical convergence and its applications to Korovkin-type approximation theorem. Int. J. Gen. Syst.
**2021**, 50, 567–579. [Google Scholar] [CrossRef] - Saini, K.; Raj, K. Applications of statistical convergence in complex uncertain sequences via deferred Riesz mean. Int. J. Uncertain. Fuzziness-Knowl.-Based Syst.
**2021**, 29, 337–351. [Google Scholar] [CrossRef] - Sharma, S.; Singh, U.P.; Raj, K. Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order (ξ,ω). J. Intell. Fuzzy Syst.
**2021**, 41, 7363–7372. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K.; Misra, U.K. Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM)
**2018**, 112, 1487–1501. [Google Scholar] [CrossRef] - Parida, P.; Paikray, S.K.; Jena, B.B. Generalized deferred Cesàro equi-statistical convergence and analogous approximation theorems. Proyecciones J. Math.
**2020**, 39, 307–331. [Google Scholar] [CrossRef] - Demirci, K.; Dirik, F.; Yıldız, S. Approximation via equi-statistical convergence in the sense of power series method. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.
**2022**, 116, 65. [Google Scholar] [CrossRef] - Ünver, M.; Orhan, C. Statistical convergence with respect to power series methods and applications to approximation theory. Numer. Funct. Anal. Optim.
**2019**, 40, 535–547. [Google Scholar] [CrossRef] - Korovkin, P.P. Convergence of linear positive operators in the spaces of continuous functions. Doklady Akad. Nauk. SSSR
**1953**, 90, 961–964. (In Russian) [Google Scholar] - Altomare, F.; Campiti, M. Korovkin-type Approximation Theory and Its Applications. In De Gruyter Studies in Mathematics; Walter de Gruyter: Berlin, Germany, 1994; Volume 17. [Google Scholar]
- Demirci, K.; Orhan, S.; Kolay, B. Statistical relative A-summation process for double sequences on modular spaces. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.
**2018**, 112, 1249–1264. [Google Scholar] [CrossRef] - Jena, B.B.; Paikray, S.K. Product of statistical probability convergence and its applications to Korovkin-type theorem. Miskolc Math. Notes
**2019**, 20, 969–984. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Deferred Cesàro statistical convergence of martingale sequence and Korovkin-type approximation theorems. Miskolc Math. Notes
**2022**, 23, 443–456. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical product convergence of martingale sequences and its applications to Korovkin-type approximation theorems. Math. Methods Appl. Sci.
**2021**, 44, 9600–9610. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical Riemann and Lebesgue integrable sequence of functions with Korovkin-type approximation theorems. Axioms
**2021**, 10, 229. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical deferred Nörlund summability and Korovkin-type approximation theorem. Mathematics
**2020**, 8, 636. [Google Scholar] [CrossRef] - Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Deferred Cesàro statistical probability convergence and its applications to approximation theorems. J. Nonlinear Convex Anal.
**2019**, 20, 1777–1792. [Google Scholar] - Balcerzak, M.; Dems, K.; Komisarski, A. Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl.
**2007**, 328, 715–729. [Google Scholar] [CrossRef] - Paikray, S.K.; Parida, P.; Mohiuddine, S.A. A certain class of relatively equi-statistical fuzzy approximation theorems. Eur. J. Pure Appl. Math.
**2020**, 13, 1212–1230. [Google Scholar] [CrossRef] - Akdağ, S. Weighted equi-statistical convergence of the Korovkin-type approximation theorems. Results Math.
**2017**, 72, 1073–1085. [Google Scholar] [CrossRef] - Dirik, F.; Demirci, K. Equi-ideal convergence of positive linear operators for analytic p-ideals. Math. Commun.
**2011**, 16, 169–178. [Google Scholar] - Karakuş, S.; Demirci, K.; Duman, O. Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl.
**2008**, 339, 1065–1072. [Google Scholar] [CrossRef] - Mohiuddine, S.A.; Alamri, B.A. Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.
**2019**, 113, 1955–1973. [Google Scholar] [CrossRef]

**Figure 1.**Equi-stat convergence of ${\mathfrak{L}}_{1}({f}_{i};x)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(i=0,1,2)$.

**Figure 2.**Equi-stat convergence of ${\mathfrak{L}}_{4}({f}_{i};x)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(i=0,1,2)$.

**Figure 3.**Equi-stat convergence of ${\mathfrak{L}}_{9}({f}_{i};x)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(i=0,1,2)$.

**Figure 4.**Equi-stat convergence of ${\mathfrak{L}}_{16}({f}_{i};x)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(i=0,1,2)$.

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**MDPI and ACS Style**

Srivastava, H.M.; Jena, B.B.; Paikray, S.K.
A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method. *Axioms* **2023**, *12*, 964.
https://doi.org/10.3390/axioms12100964

**AMA Style**

Srivastava HM, Jena BB, Paikray SK.
A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method. *Axioms*. 2023; 12(10):964.
https://doi.org/10.3390/axioms12100964

**Chicago/Turabian Style**

Srivastava, Hari Mohan, Bidu Bhusan Jena, and Susanta Kumar Paikray.
2023. "A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method" *Axioms* 12, no. 10: 964.
https://doi.org/10.3390/axioms12100964