Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems
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Oct 27, 2020
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Abstract
In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the idea of statistical convergence for sequences of real numbers, which are defined over a Banach space via the product of deferred Cesàro and deferred weighted summability means. We first establish a theorem presenting aconnection between them. Based upon our proposed methods, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space, and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as in statistical versions). Furthermore, an illustrative example is presented here by means of the generalized Meyer–König 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. Finally, we estimate the rate of the product of deferred Cesàro and deferred weighted statistical probability convergence, and accordingly establish a new result.
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References
[1] Fast H. Sur la convergence statistique. Colloquium Mathematicae, 2. 3-4:241-244. 1951.
[2] Steinhaus H. Sur la convergence ordinaire et la convergence asymp totique. Colloquium Mathematicae, 2. 1:73-74. 1951.
[3] Zygmund A. Trigonometric series. Cambridge University press. 2002.
[4] Shang Y. Estrada and L-Estrada indices of Edge-Independent random graphs. Symmetry, 7.3:1455-1462. 2015.
doi: 10.3390/sym7031455
[5] Shang Y. Estrada index of random bipartite graphs. Symmetry, 7.4:2195-2205. 2015.
doi: 10.3390/sym7042195
[6] Jena BB and Paikray SK. Product of statistical probability convergence and its applications to Korovkin-type theorem. Miskolc Mathematical Notes, 20.2:969-984. 2019.
doi: 10.18514/MMN.2019.3014
[7] Jena B, Paikray S, and Misra U. Inclusion theorems on general convergence and statistical convergence of-summability using generalized Tauberian conditions. Tamsui Oxford Journal of Information and Mathematical Sciences, 31:101-115. 2017.
[8] Jena B, Paikray S, Mohiuddine S, and Mishra VN. Relatively equi-statistical convergence via deferred Nörlund mean
based on difference operator of fractional-order and related approximation theorems. AIMS Mathematics, 5.1:650-672. 2020.
doi: 10.3934/math.2020044
[9] Kadak U, Braha N, and Srivastava H. Statistical weighted B-summability and its applications to approximation theorems.
Applied Mathematics and Computation, 302:80-96. 2017.
doi: 10.1016/j.amc.2017.01.011
[10] Mishra VN, Khatri K, Mishra LN, et al. Trigonometric approximation of periodic signals belonging to generalized
weighted Lipschitz Nörlund-Euler (N, Pn)(E, q) operator of conjugate series of its Fourier series. Journal of Classical Analysis
5.2:91-105. 2014.
doi: 10.7153/jca-05-08
[11] Mishra LN, Mishra VN, Khatri K and Deepmala. On the trigonometric approximation of signals belonging to generalized weighted Lipschitz W (Lr, ξ(t))(r ≥ 1)-class by matrix (C1 • Np) operator of conjugate series of its Fourier series. Applied Mathematics and Computation, 237:252-263. 2014.
doi: 10.1016/j.amc.2014.03.085
[12] Savaş E and Gürdal M. Ideal convergent function sequences in random 2-normed spaces. Filomat, 30.3:557-567. 2016.
doi: 10.2298/FIL1603557S
[13] Savaş E and Gürdal M. I-statistical convergence in probabilistic normed spaces. Scientific Bulletin-Series A Applied Mathematics and Physics, 77.4:195-204. 2015.
[14] Srivastava H and Et M. Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat,
31.6:1573-1582. 2017.
doi: 10.2298/FIL1706573S
[15] Zraiqat A, Paikray S, and Dutta H. A certain class of deferred weighted statistical B-summability involving (p, q) integers and analogous approximation theorems. Filomat, 33.5:1425-1444. 2019.
doi: 10.2298/FIL1905425Z
[16] Móricz F. Tauberian conditions, under which statistical convergence follows from statistical summability (C,1). Journal
of Mathematical Analysis and Applications, 275.1:277-287. 2002.
doi: 10.1016/S0022-247X(02)00338-4
[17] Mohiuddine SA, Alotaibi A, and Mursaleen M. Statistical summability (C,1) and a Korovkin type approximation theorem. Journal of Inequalities and Applications, 2012.1:172. 2012.
doi: 10.1186/1029-242X-2012-172
[18] Karakaya V and Chishti T. Weighted statistical convergence. Iranian Journal of Science and Technology Transaction A-Science, 2009.
doi: 10.22099/IJSTS.2009.2217
[19] Mursaleen M, Karakaya V, Ertürk M, and Gürsoy F. Weighted statistical convergence and its application to Korovkin type approximation theorem. Applied Mathematics and Computation, 218.18:9132-9137. 2012.
doi: 10.1016/j.amc.2012.02.068
[20] Jena BB, Paikray SK, and Misra U. Statistical deferred Cesàro summability and its applications to approximation theorems. Filomat, 32.6:2307-2319. 2018.
doi: 10.2298/FIL1806307J
[21] Srivastava H, Jena BB, Paikray SK, and Misra U. A certain class of weighted statistical convergence and associated Korovkintype approximation theorems involving trigonometric functions. Mathematical Methods in the Applied Sciences, 41.2:671-683. 2018.
doi: 10.1002/mma.4636
[22] Srivastava H, Jena BB, Paikray SK, and Misra U. Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 112.4:1487-1501. 2018.
doi: 10.1007/s13398-017-0442-3
[23] Jena BB, Paikray SK, and Dutta H. On various new concepts of statistical convergence for sequences of random variables via deferred Cesàro mean. Journal of Mathematical Analysis and Applications, 487.1:123950. 2020.
doi: 10.1016/j.jmaa.2020.123950
[24] Paikray S and Dutta H. On statistical deferred weighted B-convergence. Applied Mathematical Analysis: Theory, Methods, and Applications; Dutta, H., Peters, JF, Eds: 655-678. 2019.
doi: 10.1007/978-3-319-99918-0_20
[25] Pradhan T, Paikray S, Jena B, and Dutta H. Statistical deferred weighted B-summability and its applications to associated approximation theorems. Journal of Inequalities and Applications, 2018.1:1-21. 2018.
doi: 10.1186/s13660-018-1650-x
[26] Srivastava HM, Jena BB, and Paikray SK. Statistical Deferred Nörlund Summability and Korovkin-Type Approximation
Theorem. Mathematics, 8.4:636. 2020.
doi: 10.3390/math8040636
[27] Srivastava HM, Jena BB, Paikray SK, and Misra U. Statistically and relatively modular deferred-weighted summability and Korovkin-Type approximation theorems. Symmetry, 11.4:448. 2019.
doi: 10.3390/sym11040448
[28] Srivastava H, Jena BB, Paikray SK, and Misra U. Deferred weighted A-statistical convergence based upon the (p,
q)-Lagrange polyno- mials and its applications to approximation theorems. Journal of Applied Analysis, 24.1:1-16. 2018.
doi: 10.1515/jaa-2018-0001
[29] Srivastava HM, Jena BB, and Paikray SK. Deferred Cesàro statistical probability convergence and its applications to
approximation theorems. Journal of Nonlinear and Convex Analysis, 20.9:1777-1792. 2019.
doi: 2/jncav20-9.html
[30] Paikray S, Jena B, and Misra U. Statistical deferred Cesàro summability mean based on (p, q)-integers with application
to approximation theorems. Advances in Summability and Approximation Theory. Springer: 203-222. 2018.
doi: 10.1007/978-981-13-3077-3_13
[31] Dutta H, Paikray S, and Jena B. On statistical deferred Cesàro summability. Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Springer: 885-909. 2019.
doi: 10.1007/978-3-030-15242-0_23
[32] Agnew RP. On deferred Cesàro means. Annals of Mathematics: 413-421. 1932.
doi: 10.2307/1968524
[33] Al-Salam WA. Operational representations for the Laguerre and other polynomials. Duke Mathematical Journal, 31.1:127-142. 1964.
doi: 10.1215/S0012-7094-64-03113-8
[34] Viskov OV and Srivastava H. New Approaches to Certain Identities Involving Differential Operators. Journal of Mathematical Analysis and Applications, 186:1-10. 1994.
doi: 10.1006/jmaa.1994.1281
[35] Srivastava H. A note on certain operational representations for the Laguerre polynomials. Journal of Mathematical Analysis and Applications, 138.1:209-213. 1989.
doi: 10.1016/0022-247X(89)90331-4
[36] Srivastava H and Manocha HL. Treatise on generating functions. John Wiley & Sons. 1984.
doi: 10.1137/1028045
[37] Altın A, Doğru O, and Taşdelen F. The generalization of Meyer- König and Zeller operators by generating functions. Journal of Mathematical Analysis and Applications, 312.1:181-194. 2005.
doi: 10.1016/j.jmaa.2005.03.086
[38] Korovkin PP. Convergence of linear positive operators in the spaces of continuous functions (in Russian). Doklady Akad. Nauk. SSSR (New Ser.) 90:961-964. 1953.
[2] Steinhaus H. Sur la convergence ordinaire et la convergence asymp totique. Colloquium Mathematicae, 2. 1:73-74. 1951.
[3] Zygmund A. Trigonometric series. Cambridge University press. 2002.
[4] Shang Y. Estrada and L-Estrada indices of Edge-Independent random graphs. Symmetry, 7.3:1455-1462. 2015.
doi: 10.3390/sym7031455
[5] Shang Y. Estrada index of random bipartite graphs. Symmetry, 7.4:2195-2205. 2015.
doi: 10.3390/sym7042195
[6] Jena BB and Paikray SK. Product of statistical probability convergence and its applications to Korovkin-type theorem. Miskolc Mathematical Notes, 20.2:969-984. 2019.
doi: 10.18514/MMN.2019.3014
[7] Jena B, Paikray S, and Misra U. Inclusion theorems on general convergence and statistical convergence of-summability using generalized Tauberian conditions. Tamsui Oxford Journal of Information and Mathematical Sciences, 31:101-115. 2017.
[8] Jena B, Paikray S, Mohiuddine S, and Mishra VN. Relatively equi-statistical convergence via deferred Nörlund mean
based on difference operator of fractional-order and related approximation theorems. AIMS Mathematics, 5.1:650-672. 2020.
doi: 10.3934/math.2020044
[9] Kadak U, Braha N, and Srivastava H. Statistical weighted B-summability and its applications to approximation theorems.
Applied Mathematics and Computation, 302:80-96. 2017.
doi: 10.1016/j.amc.2017.01.011
[10] Mishra VN, Khatri K, Mishra LN, et al. Trigonometric approximation of periodic signals belonging to generalized
weighted Lipschitz Nörlund-Euler (N, Pn)(E, q) operator of conjugate series of its Fourier series. Journal of Classical Analysis
5.2:91-105. 2014.
doi: 10.7153/jca-05-08
[11] Mishra LN, Mishra VN, Khatri K and Deepmala. On the trigonometric approximation of signals belonging to generalized weighted Lipschitz W (Lr, ξ(t))(r ≥ 1)-class by matrix (C1 • Np) operator of conjugate series of its Fourier series. Applied Mathematics and Computation, 237:252-263. 2014.
doi: 10.1016/j.amc.2014.03.085
[12] Savaş E and Gürdal M. Ideal convergent function sequences in random 2-normed spaces. Filomat, 30.3:557-567. 2016.
doi: 10.2298/FIL1603557S
[13] Savaş E and Gürdal M. I-statistical convergence in probabilistic normed spaces. Scientific Bulletin-Series A Applied Mathematics and Physics, 77.4:195-204. 2015.
[14] Srivastava H and Et M. Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat,
31.6:1573-1582. 2017.
doi: 10.2298/FIL1706573S
[15] Zraiqat A, Paikray S, and Dutta H. A certain class of deferred weighted statistical B-summability involving (p, q) integers and analogous approximation theorems. Filomat, 33.5:1425-1444. 2019.
doi: 10.2298/FIL1905425Z
[16] Móricz F. Tauberian conditions, under which statistical convergence follows from statistical summability (C,1). Journal
of Mathematical Analysis and Applications, 275.1:277-287. 2002.
doi: 10.1016/S0022-247X(02)00338-4
[17] Mohiuddine SA, Alotaibi A, and Mursaleen M. Statistical summability (C,1) and a Korovkin type approximation theorem. Journal of Inequalities and Applications, 2012.1:172. 2012.
doi: 10.1186/1029-242X-2012-172
[18] Karakaya V and Chishti T. Weighted statistical convergence. Iranian Journal of Science and Technology Transaction A-Science, 2009.
doi: 10.22099/IJSTS.2009.2217
[19] Mursaleen M, Karakaya V, Ertürk M, and Gürsoy F. Weighted statistical convergence and its application to Korovkin type approximation theorem. Applied Mathematics and Computation, 218.18:9132-9137. 2012.
doi: 10.1016/j.amc.2012.02.068
[20] Jena BB, Paikray SK, and Misra U. Statistical deferred Cesàro summability and its applications to approximation theorems. Filomat, 32.6:2307-2319. 2018.
doi: 10.2298/FIL1806307J
[21] Srivastava H, Jena BB, Paikray SK, and Misra U. A certain class of weighted statistical convergence and associated Korovkintype approximation theorems involving trigonometric functions. Mathematical Methods in the Applied Sciences, 41.2:671-683. 2018.
doi: 10.1002/mma.4636
[22] Srivastava H, Jena BB, Paikray SK, and Misra U. Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 112.4:1487-1501. 2018.
doi: 10.1007/s13398-017-0442-3
[23] Jena BB, Paikray SK, and Dutta H. On various new concepts of statistical convergence for sequences of random variables via deferred Cesàro mean. Journal of Mathematical Analysis and Applications, 487.1:123950. 2020.
doi: 10.1016/j.jmaa.2020.123950
[24] Paikray S and Dutta H. On statistical deferred weighted B-convergence. Applied Mathematical Analysis: Theory, Methods, and Applications; Dutta, H., Peters, JF, Eds: 655-678. 2019.
doi: 10.1007/978-3-319-99918-0_20
[25] Pradhan T, Paikray S, Jena B, and Dutta H. Statistical deferred weighted B-summability and its applications to associated approximation theorems. Journal of Inequalities and Applications, 2018.1:1-21. 2018.
doi: 10.1186/s13660-018-1650-x
[26] Srivastava HM, Jena BB, and Paikray SK. Statistical Deferred Nörlund Summability and Korovkin-Type Approximation
Theorem. Mathematics, 8.4:636. 2020.
doi: 10.3390/math8040636
[27] Srivastava HM, Jena BB, Paikray SK, and Misra U. Statistically and relatively modular deferred-weighted summability and Korovkin-Type approximation theorems. Symmetry, 11.4:448. 2019.
doi: 10.3390/sym11040448
[28] Srivastava H, Jena BB, Paikray SK, and Misra U. Deferred weighted A-statistical convergence based upon the (p,
q)-Lagrange polyno- mials and its applications to approximation theorems. Journal of Applied Analysis, 24.1:1-16. 2018.
doi: 10.1515/jaa-2018-0001
[29] Srivastava HM, Jena BB, and Paikray SK. Deferred Cesàro statistical probability convergence and its applications to
approximation theorems. Journal of Nonlinear and Convex Analysis, 20.9:1777-1792. 2019.
doi: 2/jncav20-9.html
[30] Paikray S, Jena B, and Misra U. Statistical deferred Cesàro summability mean based on (p, q)-integers with application
to approximation theorems. Advances in Summability and Approximation Theory. Springer: 203-222. 2018.
doi: 10.1007/978-981-13-3077-3_13
[31] Dutta H, Paikray S, and Jena B. On statistical deferred Cesàro summability. Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Springer: 885-909. 2019.
doi: 10.1007/978-3-030-15242-0_23
[32] Agnew RP. On deferred Cesàro means. Annals of Mathematics: 413-421. 1932.
doi: 10.2307/1968524
[33] Al-Salam WA. Operational representations for the Laguerre and other polynomials. Duke Mathematical Journal, 31.1:127-142. 1964.
doi: 10.1215/S0012-7094-64-03113-8
[34] Viskov OV and Srivastava H. New Approaches to Certain Identities Involving Differential Operators. Journal of Mathematical Analysis and Applications, 186:1-10. 1994.
doi: 10.1006/jmaa.1994.1281
[35] Srivastava H. A note on certain operational representations for the Laguerre polynomials. Journal of Mathematical Analysis and Applications, 138.1:209-213. 1989.
doi: 10.1016/0022-247X(89)90331-4
[36] Srivastava H and Manocha HL. Treatise on generating functions. John Wiley & Sons. 1984.
doi: 10.1137/1028045
[37] Altın A, Doğru O, and Taşdelen F. The generalization of Meyer- König and Zeller operators by generating functions. Journal of Mathematical Analysis and Applications, 312.1:181-194. 2005.
doi: 10.1016/j.jmaa.2005.03.086
[38] Korovkin PP. Convergence of linear positive operators in the spaces of continuous functions (in Russian). Doklady Akad. Nauk. SSSR (New Ser.) 90:961-964. 1953.
How to Cite
Jena, B. B., & Paikray, S. K. (2020). Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems. Universitas Scientiarum, 25(3), 409–433. https://doi.org/10.11144/Javeriana.SC25-3.podc
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