Approximation on Modular Spaces via P-Statistical Relative A-Summation Process

Öz

In this paper, we first present the notions of statistical relative modular and F-norm convergence concerning the power series method. Then, we also present theorems of Korovkin-type via statistical relative A-summation process via power series method on modular spaces, including as particular cases weighted spaces, certain interpolation spaces, Orlicz and Musielak-Orlicz spaces, Lp spaces and many others. Later, we consider some application to Kantorovich-type operators in Orlicz spaces. Moreover, we present some estimates of rates of convergence via modulus of continuity. We end the paper with giving some concluding remarks.

Anahtar Kelimeler

• Korovkin theorem
• modular space
• statistical convergence
• matrix summability
• power series method

Modüler Uzaylar Üzerinde P-İstatistiksel A-Toplam Süreci Aracılığıyla Yaklaşım

Öz

Bu çalışmada, ilk olarak, kuvvet serisi yöntemiyle ilgili istatistiksel relative modüler ve F-norm yakınsama kavramlarını sunuyoruz. Daha sonra, özel durumlar olarak ağırlıklı uzaylar, belirli enterpolasyon uzayları, Orlicz ve Musielak-Orlicz uzayları, Lp uzayları ve diğer birçok uzayları içeren modüler uzaylar üzerinde kuvvet serisi yöntemiyle istatistiksel relative A-toplam süreci aracılığıyla Korovkin-tipi teoremleri de sunuyoruz. Daha sonra, Orlicz uzaylarında Kantorovich tipi operatörlere bazı uygulamaları göz önüne alıyoruz. Dahası, süreklilik modülü aracılığıyla yakınsama oranlarının bazı tahminlerini sunuyoruz. Makaleyi bazı son sözler vererek bitiriyoruz.

Anahtar Kelimeler

• Korovkin teoremi
• modüler uzay
• istatistiksel yakınsaklık
• matris toplanabilme
• kuvvet serisi yöntemi

Kaynakça

1. Niven, I., Zuckerman, H. S., & Montgomery, H. L. (1991). An Introduction to the theory of numbers. John Wiley & Sons.
2. Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicae, 2(3-4), 241-244.
3. Steinhaus, H. (1951). Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicae, 2(1), 73-74.
4. Ünver, M., & Orhan, C. (2019). Statistical convergence with respect to power series methods and applications to approximation theory. Numerical Functional Analysis and Optimization, 40(5), 535-547. https://doi.org/10.1080/01630563.2018.1561467
5. Kratz, W., & Stadtmüller, U. (1989). Tauberian theorems for Jp-summability. Journal of Mathematical Analysis and Applications, 139(2), 362-371. https://doi.org/10.1016/0022-247X(89)90113-3
6. Stadtmüller, U., & Tali, A. (1999). On certain families of generalized Nörlund methods and power series methods. Journal of Mathematical Analysis and Applications, 238(1), 44-66. https://doi.org/10.1006/jmaa.1999.6503
7. Boos, J. (2000). Classical and modern methods in summability. Oxford University Press.
8. Fridy, J., & Orhan, C. (1997). Statistical limit superior and limit inferior. Proceedings of the American Mathematical Society, 125(12), 3625-3631. https://doi.org/10.1090/S0002-9939-97-04000-8

9. Demirci, K. (2001). I-limit superior and limit inferior. Mathematical Communications, 6(2), 165-172.
10. Bardaro, C., Musielak, J., & Vinti, G. (2003). Nonlinear integral operators and applications. Volume 9 in the series De Gruyter Series in Nonlinear Analysis and Applications.
11. Musielak, J. (1993). Nonlinear approximation in some modular function spaces I. Mathematica Japonica, 38(1), 83-90.
12. Kozlowski, W. M. (1988). Modular function spaces. Pure Appl. Math. 122 Marcel Dekker, Inc.
13. Musielak, J. (1983). Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics Vol. 1034 Springer-Verlag.
14. Yılmaz, B., Demirci, K., & Orhan, S. (2016). Relative modular convergence of positive linear operators. Positivity, 20(3), 565-577. https://doi.org/10.1007/s11117-015-0372-2
15. Bell, H. T. (1973). Order summability and almost convergence. Proceedings of the American Mathematical Society, 38(3), 548-552. https://doi.org/10.1090/S0002-9939-1973-0310489-8
16. Stieglitz, M. (1973). Eine verallgemeinerung des begriffs der fastkonvergenz. Math. Japon, 18(1), 53-70.
17. Orhan, S., & Demirci, K. (2014). Statistical A-summation process and Korovkin type approximation theorem on modular spaces. Positivity, 18(4), 669-686. https://doi.org/10.1007/s11117-013-0269-x
18. Kolay, B., Orhan, S., & Demirci, K. (2018). Statistical Relative A-Summation Process and Korovkin-Type Approximation Theorem on Modular Spaces. Iranian Journal of Science and Technology, Transactions A: Science, 42(2), 683-692. https://doi.org/10.1007/s40995-016-0137-1
19. Ditzian, Z. & Totik, V. (1987). Moduli of smoothness. Springer series in computational mathematics, Springer-Verlag.
20. Lorentz, G. G. (1948). A contribution to the theory of divergent sequences. Acta Mathematica, 80(1), 167-190. https://doi.org/10.1007/BF02393648
21. Bardaro, C. & Mantellini, I. (2007). Korovkin’s theorem in modular spaces. Commentationes Mathematicae, 47(2), 239-253.
22. Bardaro, C., & Mantellini, I. (2006). Approximation properties in abstract modular spaces for a class of general sampling-type operators. Applicable Analysis, 85(4), 383-413. https://doi.org/10.1080/00036810500380332
23. Mantellini, I. (1998). Generalized sampling operators in modular spaces. In Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae, 38, 77-92.
24. Devore, R. A. (1972). The approximation of continuous functions by positive linear operators. Lecture Notes in Mathematics, 14 Springer-Verlag, Vol. 293.
25. Korovkin, P. P. (1960). Linear operators and approximation theory. New Delhi, India: Hindustan Publishing Company.
26. Altomare, F. (2010). Korovkin-type Theorems and Approximation by Positive Linear Operators. Surveys in Approximation Theory, 5, 92-164. http://arxiv.org/abs/1009.2601
27. Ansari, K. J., Özger, F., & Ödemi¸s Özger, Z. (2022). Numerical and theoretical approximation results for Schurer–Stancu operators with shape parameter λ. Computational and Applied Mathematics, 41(4), 1-18. https://doi.org/10.1007/s40314-022-01877-4
28. Bardaro, C., Boccuto, A., Demirci, K., Mantellini, I., & Orhan, S. (2015). Triangular A-statistical approximation by double sequences of positive linear operators. Results in Mathematics, 68(1), 271-291. https://doi.org/10.1007/s00025-015-0433-7
29. Bardaro, C., Boccuto, A., Demirci, K., Mantellini, I., & Orhan, S. (2015). Korovkin-Type Theorems for Modular Ψ-A-Statistical Convergence. Journal of Function Spaces, 2015, Article ID 160401. https://doi.org/10.1155/2015/160401
30. Bayram, N. ¸ S. (2016). Strong summation process in locally integrable function spaces. Hacettepe Journal of Mathematics and Statistics, 45(3), 683-694.
31. Bayram, N. ¸ S., & Orhan, C. (2020). A-Summation process in the space of locally integrable functions. Studia Universitatis Babes-Bolyai, Mathematica, 65(2), 255-268. https://doi.org/10.24193/subbmath.2020.2.07
32. Cai, Q. B., Ansari, K. J., Temizer Ersoy, M., & Özger, F. (2022). Statistical blending-type approximation by a class of operators that includes shape parameters λ and α. Mathematics, 10(7), 1149.
33. Demirci, K., Dirik, F., & Yıldız, S. (2022). Approximation via equi-statistical convergence in the sense of power series method. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 116(2), 1-13. https://doi.org/10.1007/s13398-021-01191-4
34. Demirci, K., & Dirik, F. (2011). Approximation for periodic functions via statistical σ-convergence. Mathematical Communications, 16(1), 77-84.
35. Demirci, K., & Orhan, S. (2017). Statistical relative approximation on modular spaces. Results in Mathematics, 71(3), 1167-1184. https://doi.org/10.1007/s00025-016-0548-5
36. Demirci, K., & Dirik, F. (2010). Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions. Mathematical and Computer Modelling, 52(9-10), 1858-1866. https://doi.org/10.1016/j.mcm.2010.07.015
37. Demirci, K., Yıldız, S., & Dirik, F. (2020). Approximation via power series method in two-dimensional weighted spaces. Bulletin of the Malaysian Mathematical Sciences Society, 43(6), 3871-3883. https://doi.org/10.1007/s40840-020-00902-1
38. Gadjiev, A. D., & Orhan, C. (2002). Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics, 32(1), 129-138.
39. Sahin, P. O., & Dirik, F. (2017). Statistical relative uniform convergence of double sequences of positive linear operators. Applied Mathematics E-Notes, 17, 207-220.
40. Orhan, S., & Demirci, K. (2015). Statistical approximation by double sequences of positive linear operators on modular spaces. Positivity, 19(1), 23-36. https://doi.org/10.1007/s11117-014-0280-x
41. Özger, F., Aljimi, E., & Temizer Ersoy, M. (2022). Rate of weighted statistical convergence for generalized blending-type Bernstein-Kantorovich operators. Mathematics, 10(12), 2027. https://doi.org/10.3390/math10122027
42. Karakuş, S., Demirci, K., & Duman, O. (2010). Statistical approximation by positive linear operators on modular spaces. Positivity, 14(2), 321-334. https://doi.org/10.1007/s11117-009-0020-9
43. Karakuş, S., & Demirci, K. (2010). Matrix summability and Korovkin type approximation theorem on modular spaces. Acta Mathematica Universitatis Comenianae, 79(2), 281-292.
44. Nishishiraho, T. (1981). Quantitative theorems on linear approximation processes of convolution operators in Banach spaces. Tohoku Mathematical Journal, Second Series, 33(1), 109-126. https://doi.org/10.2748/tmj/1178229498
45. Nishishiraho, T. (1983). Convergence of positive linear approximation processes. Tohoku Mathematical Journal, Second Series, 35(3), 441-458. https://doi.org/10.2748/tmj/1178229002
46. Kolk, E. (1993). Matrix summability of statistically convergent sequences. Analysis, 13(1-2), 77-84. https://doi.org/10.1524/anly.1993.13.12.77
47. Altomare, F., & Campiti, M. (1994). Korovkin-type approximation theory and its Applications. Volume 17 in the series De Gruyter Studies in Mathematics.