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Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Nörlund summability mean. 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 random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by 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.

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References

  1. Agnew, R.P.: On deferred Cesàro means. Ann. Math. 33, 413–421 (1932)

    MathSciNet  MATH  Google Scholar 

  2. Altın, A., Doǧru, O., Taşdelen, F.: The generalization of Meyer-König and Zeller operators by generating functions. J. Math. Anal. Appl. 312, 181–194 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Al-Salam, W.A.: Operational representations for the Laguerre and other polynomials. Duke Math. J. 31, 127–142 (1964)

    MathSciNet  MATH  Google Scholar 

  4. Braha, N.L., Loku, V., Srivastava, H.M.: \(\Lambda ^{2}\)-Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Appl. Math. Comput. 266, 675–686 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Braha, N.L., Srivastava, H.M., Mohiuddine, S.A.: A Korovkin-type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Das, P., Ghosal, S., Som, S.: Statistical convergence of order \(\alpha \) in probability. Arab J. Math. Sci. 21, 253–265 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Das, A.A., Jena, B.B., Paikray, S.K., Jati, R.K.: Statistical deferred weighted summability and associated Korovokin-type approximation theorem. Nonlinear Sci. Lett. A 9, 238–245 (2018)

    Google Scholar 

  8. Das, A.A., Paikray, S.K., Pradhan, T., Dutta, H.: Statistical \((C,1)(E,\mu )\)-summablity and associated fuzzy approximation theorems with statistical fuzzy rates. Soft Comput. 1–10, (2019)

  9. Das, A.A., Paikray, S.K., Pradhan, T., Dutta, H.: Approximation of signals in the weighted Zygmund class via Euler-Hausdorff product summability mean of Fourier series. J. Indian Math. Soc. (New Ser.) 86, 296–314 (2019)

    Google Scholar 

  10. Dutta, H., Paikray, S.K., Jena, B.B.: On statistical deferred Cesàro summability. In: Dutta, H., Ljubiša Kočinac, D.R., Srivastava, H.M. (eds.) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications, pp. 885–909. Springer Nature Switzerland AG, Cham (2019)

    Google Scholar 

  11. Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)

    MathSciNet  MATH  Google Scholar 

  12. Jena, B.B., Paikray, S.K.: Product of statistical probability convergence and its applications to Korovkin-type theorem. Miskolc Math. Notes 20, 969–984 (2019)

    MATH  Google Scholar 

  13. Jena, B.B., Paikray, S.K., Dutta, H.: On various new concepts of statistical convergence for sequences of random variables via deferred Cesàro mean. J. Math. Anal. Appl. 487, 1–18 (2020). Article ID 123950

    MATH  Google Scholar 

  14. Jena, B.B., Paikray, S.K., Misra, U.K.: Approximation of periodic functions via statistical \(B\)-summability and its applications to approximation theorems. Indian J. Ind. Appl. Math. 10, 71–86 (2019)

    Google Scholar 

  15. Jena, B.B., Paikray, S.K., Misra, U.K.: Statistical deferred Cesàro summability and its applications to approximation theorems. Filomat 32, 2307–2319 (2018)

    MathSciNet  Google Scholar 

  16. Jena, B.B., Paikray, S.K., Misra, U.K.: Inclusion theorems on general convergence and statistical convergence of \((L,1,1)\)-summability using generalized Tauberian conditions. Tamsui Oxf. J. Inf. Math. Sci. 31, 101–115 (2017)

    MathSciNet  Google Scholar 

  17. Jena, B.B., Paikray, S.K., Mohiuddine, S.A., Mishra, V.N.: Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems. AIMS Math. 5, 650–672 (2020)

    Google Scholar 

  18. Kadak, U., Braha, N.L., Srivastava, H.M.: Statistical weighted \({\cal{B}}\)-summability and its applications to approximation theorems. Appl. Math. Comput. 302, 80–96 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Karakaya, V., Chishti, T.A.: Weighted statistical convergence. Iran. J. Sci. Technol. Trans. A: Sci. 33(A3), 219–223 (2009)

    MathSciNet  Google Scholar 

  20. Korovkin, P.P.: Convergence of linear positive operators in the spaces of continuous functions. Doklady Akad. Nauk. SSSR (New Ser.) 90, 961–964 (1953). (in Russian)

    MathSciNet  Google Scholar 

  21. Mishra, L.N., Patro, M., Paikray, S.K., Jena, B.B.: A certain class of statistical deferred weighted \(A\)-summability based on \((p, q)\)-integers and associated approximation theorems. Appl. Appl. Math. 14, 716–740 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Mohiuddine, S.A.: Statistical weighted \(A\)-summability with application to Korovkin’s type approximation theorem. J. Inequal. Appl. 2016, 1–13 (2016). Article ID 101

    MathSciNet  MATH  Google Scholar 

  23. Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 113, 1955–1973 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical summability \((C,1)\) and a Korovkin-type approximation theorem. J. Inequal. Appl. 2012, 1–8 (2012). Article ID 172

    MathSciNet  MATH  Google Scholar 

  25. Móricz, F.: Tauberian conditions under which statistical convergence follows from statistical summability \((C,1)\). J. Math. Anal. Appl. 275, 277–287 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Mursaleen, M., Karakaya, V., Ertürk, M., Gürsoy, F.: Weighted statistical convergence and its application to Korovkin-type approximation theorem. Appl. Math. Comput. 218, 9132–9137 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Paikray, S.K., Dutta, H.: On statistical deferred weighted \({\cal{B}}\)-convergence. In: Dutta, H., Peters, J.F. (eds.) Applied Mathematical Analysis: Theory, Methods, and Applications, pp. 655–678. Springer Nature Switzerland AG, Cham (2019)

    Google Scholar 

  28. Paikray, S.K., Jena, B.B., Misra, U.K.: Statistical deferred Cesàro summability mean based on \((p, q)\)-integers with application to approximation theorems. In: Mohiuddine, S.A., Acar, T. (eds.) Advances in Summability and Approximation Theory, pp. 203–222. Springer Nature Singapore Private Limited, Singapore (2019)

    Google Scholar 

  29. Patro, M., Paikray, S.K., Jena, B.B., Dutta, H.: Statistical deferred Riesz summability mean and associated approximation theorems for trigonometric functions. In: Singh, J., Kumar, D., Dutta, H., Baleanu, D., Purohit, S.D. (eds.) Mathematical Modeling, Applied Analysis and Computation, pp. 53–67. Springer Nature Singapore Private Limited, Singapore (2019)

    MATH  Google Scholar 

  30. Pradhan, T., Jena, B.B., Paikray, S.K., Dutta, H., Misra, U.K.: On approximation of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Nörlund mean. Afrika Mat. 30, 1119–1131 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Pradhan, T., Paikray, S.K., Das, A.A., Dutta, H.: On approximation of signals in the generalized Zygmund class via \((E,1)(N, p_n)\) summability means of conjugate Fourier series. Proyecciones J. Math. 38, 1015–1033 (2019)

    Google Scholar 

  32. Pradhan, T., Paikray, S.K., Jena, B.B., Dutta, H.: Statistical deferred weighted \({\cal{B}}\)-summability and its applications to associated approximation theorems. J. Inequal. Appl. 2018, 1–21 (2018). Article ID 65

    MathSciNet  Google Scholar 

  33. Şençimen, C.: Statistical convergence in probability for a sequence of random functions. J. Theoret. Probab. 26, 94–106 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Srivastava, H.M.: A note on certain operational representations for the Laguerre polynomials. J. Math. Anal. Appl. 138, 209–213 (1989)

    MathSciNet  MATH  Google Scholar 

  35. Srivastava, H.M., Et, M.: Lacunary statistical convergence and strongly lacunary summable functions of order \(\alpha \). Filomat 31, 1573–1582 (2017)

    MathSciNet  Google Scholar 

  36. Srivastava, H.M., Jena, B.B., Paikray, S.K.: A certain class of statistical probability convergence and its applications to approximation theorems, Appl. Anal. Discrete Math. (in press) (2020)

  37. Srivastava, H.M., Jena, B.B., Paikray, S.K.: Statistical deferred Nörlund summability and Korovkin-type approximation theorem. Mathematics 8, 1–11 (2020). Article ID 636

    Google Scholar 

  38. Srivastava, H.M., Jena, B.B., Paikray, S.K.: Deferred Cesàro statistical probability convergence and its applications to approximation theorems. J. Nonlinear Convex Anal. 20, 1777–1792 (2019)

    MathSciNet  Google Scholar 

  39. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Statistically and relatively modular deferred-weighted summability and Korovkin-type approximation theorems. Symmetry 11, 1–20 (2019). Article ID 448

    MATH  Google Scholar 

  40. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: A certain class of weighted statistical convergence and associated Korovkin type approximation theorems for trigonometric functions. Math. Methods Appl. Sci. 41, 671–683 (2018)

    MathSciNet  MATH  Google Scholar 

  41. 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) 112, 1487–1501 (2018)

    MathSciNet  MATH  Google Scholar 

  42. Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K.: Deferred weighted \({\cal{A}}\)-statistical convergence based upon the \((p, q)\)-Lagrange polynomials and its applications to approximation theorems. J. Appl. Anal. 24, 1–16 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1984)

  44. Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73–74 (1951)

    MathSciNet  Google Scholar 

  45. Viskov, O.V., Srivastava, H.M.: New approaches to certain identities involving differential operators. J. Math. Anal. Appl. 186, 1–10 (1994)

    MathSciNet  MATH  Google Scholar 

  46. Zraiqat, A., Paikray, S.K., Dutta, H.: A certain class of deferred weighted statistical \(B\)-summability involving \((p, q)\)-integers and analogous approximation theorems. Filomat 33, 1425–1444 (2019)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada

    H. M. Srivastava

  2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan, Republic of China

    H. M. Srivastava

  3. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    H. M. Srivastava

  4. Department of Mathematics, Veer Surendra Sai University of Technology, Burla, 768018, Odisha, India

    Bidu Bhusan Jena & Susanta Kumar Paikray

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  1. H. M. Srivastava
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  2. Bidu Bhusan Jena
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  3. Susanta Kumar Paikray
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Correspondence to Susanta Kumar Paikray.

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Srivastava, H.M., Jena, B.B. & Paikray, S.K. Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems. RACSAM 114, 144 (2020). https://doi.org/10.1007/s13398-020-00875-7

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