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Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems

1 Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India
2 Department of General Required Courses, Mathematics, Faculty of Applied Studies, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3 Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India

Special Issues: 2nd International Conference on Mathematical Modeling, Applied Analysis and Computation (ICMMAAC-19), August 8–10, 2019, JECRC University, Jaipur, India

In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Nörlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Nörlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.
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Keywords deferred Nörlund mean; relatively statistical uniform convergence; relatively ψnp,q-equi-statistical convergence; Korovkin-type approximation theorem; rate of the relatively ψnp,q-equi-statistical convergence

Citation: B. B. Jena, S. K. Paikray, S. A. Mohiuddine, Vishnu Narayan Mishra. Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems. AIMS Mathematics, 2020, 5(1): 650-672. doi: 10.3934/math.2020044

References

  • 1. T. Acar, A. Aral, S. A. Mohiuddine, On Kantorovich modifications of (p, q)-Baskakov operators, J. Inequal. Appl., 2016 (2016), 1-14.    
  • 2. T. Acar, S. A. Mohiuddine, Statistical (C, 1)(E, 1) summability and Korovkin's theorem, Filomat, 30 (2016), 387-393.
  • 3. R. P. Agnew, On deferred Cesàro means, Ann. Math., 33 (1932), 413-421.    
  • 4. H. Aktuǧlu, H. Gezer, Lacunary equi-statistical convergence of positive linear operators, Cent. Eur. J. Math., 7(2009), 558-567.    
  • 5. W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. J., 31 (1964), 127-142.    
  • 6. M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl., 328 (2007), 715-729.    
  • 7. P. Baliarsingh, On a fractional difference operator, Alex. Eng. J., 55 (2016), 1811-1816.    
  • 8. F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012.
  • 9. C. A. Bektaş, M. Et, R. Çolak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004), 423-432.    
  • 10. C. Belen, S. A. Mohiuddine, Generalized statistical convergence and application, Appl. Math. Comput., 219 (2013), 9821-9826.
  • 11. N. L. Braha, H. M. Srivastava, S. A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean, Appl. Math. Comput., 228 (2014), 162-169.
  • 12. S. Chapman, On non-integral orders of summability of series and integrals, Proc. Lond. Math. Soc., 2 (1911), 369-409.
  • 13. E. W. Chittenden, On the limit functions of sequences of continuous functions converging relatively uniformly, Trans. AMS, 20 (1919), 179-184.    
  • 14. A. A. Das, B. B. Jena, S. K. Paikray, et al. Statistical deferred weighted summability and associated Korovokin-type approximation theorem, Nonlinear Sci. Lett. A, 9 (2018), 238-245.
  • 15. K. Demirci, S. Orhan, Statistically relatively uniform convergence of positive linear operators, Results Math., 69 (2016), 359-367.    
  • 16. K. Demirci, S. Orhan, Statistical relative approximation on modular spaces, Results Math., 71 (2017), 1167-1184.    
  • 17. O. H. H. Edely, S. A. Mohiuddine, A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett., 23 (2010), 1382-1387.    
  • 18. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.    
  • 19. G. Gasper, M. Rahman, Basic Hypergeometric Series, Camb. Univ. Press, 2004.
  • 20. F. H. Jackson, Ageneralization of the functions Γ(n) and xn, Proc. R. Soc. Lond., 74 (1904), 64-72.    
  • 21. B. B. Jena, S. K. Paikray, Product of statistical probability convergence and its applications to Korovkin-type theorem, Miskolc Math. Notes, 20 (2019), 1-16.
  • 22. B. B. Jena, S. K. Paikray, U. K. Misra, Statistical deferred Cesàro summability and its applications to approximation theorems, Filomat, 32 (2018), 2307-2319.    
  • 23. B. B. Jena, S. K. Paikray, U. K. Misra, Inclusion theorems on general convergence and statistical convergence of (L, 1, 1)-summability using generalized Tauberian conditions, Tamsui Oxf. J. Inf. Math. Sci., 31 (2017), 101-115.
  • 24. U. Kadak, Weighted statistical convergence based on generalized difference operator involving (p, q)-gamma function and its applications to approximation theorems, J. Math. Anal. Appl., 448 (2017), 1633-1650.
  • 25. U. Kadak, On weighted statistical convergence based on (p, q)-integers and related approximation theorems for functions of two variables, J. Math. Anal. Appl., 443 (2016), 752-764.
  • 26. U. Kadak, P. Baliarsingh, On certain Euler difference sequence spaces of fractional order and related dual properties, J. Nonlinear Sci. Appl., 8 (2015), 997-1004.    
  • 27. U. Kadak, S. A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-gamma function and related approximation theorems, Results Math., 73 (2018), Article 9.
  • 28. U. Kadak, H. M. Srivastava, M. Mursaleen, Relatively uniform weighted summability based on fractional-order difference operator, Bull. Malays. Math. Sci. Soc., 42 (2019), 2453-2480.    
  • 29. V. Karakaya, T. A. Chishti, Weighted statistical convergence, Iranian. J. Sci. Technol. Trans. A, 33 (A3)(2009), 219-223.
  • 30. S. Karakuş, K. Demirci, O. Duman, Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 339 (2008), 1065-1072.    
  • 31. P. P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Co., Delhi, 1960.
  • 32. A. Lupaş, A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, 9 (1987), 85-92.
  • 33. S. A. Mohiuddine, An application of almost convergence in approximation theorems, Appl. Math. Lett., 24 (2011), 1856-1860.    
  • 34. S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl., 2016 (2016).
  • 35. S. A. Mohiuddine, B. A. S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat., RACSAM, 113(2019), 1955-1973.    
  • 36. S. A. Mohiuddine, A. Asiri, B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst., 48((2019), 492-506.
  • 37. S. A. Mohiuddine, B. Hazarika, M. A. Alghamdi, Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33((2019), 4549-4560.
  • 38. E. H. Moore, An Introduction to a Form of General Analysis, The New Haven Mathematical Colloquium, Yale University Press, New Haven 1910.
  • 39. M. Mursaleen, K. J. Ansari, A. Khan, On (p, q)-analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), 874-882.
  • 40. M. Mursaleen, V. Karakaya, M. Ertürk, et al. Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132-9137.
  • 41. M. Mursaleen, Md. Nasiuzzaman, A. Nurgali, Some approximation results on Bernstein-Schurer operators defined by (p, q)-integers, J. Inequal. Appl., 2015 (2015), 1-12.    
  • 42. T. Pradhan, S. K. Paikray, B. B. Jena, et al. Statistical deferred weighted B-summability and its applications to associated approximation theorems, J. Inequal. Appl., 2018 (2018), 1-21.    
  • 43. P. N. Sadjang, On the (p, q)-gamma and (p, q)-beta function, arXiv: 1506. 07394v1.
  • 44. H. M. Srivastava, B. B. Jena, S. K. Paikray, et al. A certain class of weighted statistical convergence and associated Korovkin type approximation theorems for trigonometric functions, Math. Methods Appl. Sci., 41 (2018), 671-683.
  • 45. H. M. Srivastava, B. B. Jena, S. K. Paikray, et al. Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 112 (2018), 1487-1501.
  • 46. H. M. Srivastava, B. B. Jena, S. K. Paikray, et al. Deferred weighted A-statistical convergence based upon the (p, q)-Lagrange polynomials and its applications to approximation theorems, J. Appl. Anal., 24 (2018), 1-16.
  • 47. H. M. Srivastava, B. B. Jena, S. K. Paikray, et al. Statistically and relatively modular deferredweighted summability and Korovkin-type approximation theorems, Symmetry, 11 (2019), 1-20.
  • 48. H. M. Srivastava, B. B. Jena, S. K. Paikray, Deferred Cesàro statistical probability convergence and its applications to approximation theorems, J. Nonlinear Convex Anal., 20 (2019), 1777-1792.
  • 49. H. M. Srivastava, B. B. Jena, S. K. Paikray, A certain class of statistical probability convergence and its applications to approximation theorems, Appl. Anal. Discrete Math. (in press) 2019.
  • 50. H. M. Srivastava, H. L. Manocha, Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  • 51. H. M. Srivastava, M. Mursaleen, A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Model. 55 (2012), 2040-2051.    
  • 52. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
  • 53. O. V. Viskov, H. M. Srivastava, New approaches to certain identities involving differential operators, J. Math. Anal. Appl., 186 (1994), 1-10.    

 

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