Generalized Cesàro summability of Fourier series and its applications

Abstract

In this paper, by using generalized Cesàro means based on q-integers we study on approximating continuous and periodic functions by its Fourier series. We also discuss its connection with the concept of statistical convergence. At the end of the paper, some applications and graphical illustrations are also provided.

Keywords

• Fourier Analysis
• Cesaro summability
• Fejer's kernel
• q-integers
• statistical convergence

References

1. H. Aktuglu, S. Bekar: q-Cesàro matrix and q-statistical convergence, J. Comput. Appl. Math., 235 (16) (2011), 4717–4723.
2. F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, (1994).
3. G. A. Anastassiou, O. Duman: Towards intelligent modeling: statistical approximation theory, Intelligent Systems Reference Library, 14. Springer-Verlag, Berlin, (2011).
4. J. Bustoz, L. F. Gordillo: q-Hausdorff summability, J. Comput. Anal. Appl., 7 (1) (2005), 35–48.
5. J. S. Connor: The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1–2) (1988), 47–63.
6. O. Duman: Statistical approximation for periodic functions, Demonstratio Math., 36 (4) (2003), 873–878.
7. O. Duman, M. K. Khan and C. Orhan: A-statistical convergence of approximating operators, Math. Inequal. Appl., 6 (4) (2003), 689–699.
8. H. Fast: Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.

9. J. A. Frid: On statistical convergence Analysis, 5 (4) (1985), 301–313.
10. A. D. Gadjiev, C. Orhan: Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (1) (2002), 129–138.
11. V. Kac, P. Cheung: Quantum calculus, Universitext. Springer-Verlag, New York, (2002).
12. P. P. Korovkin: Linear operators and approximation theory, Translated from the Russian ed. (1959). Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. III. Gordon and Breach Publishers, Inc., New York; Hindustan Publishing Corp., Delhi, India, (1960).
13. F. Móricz: Statistical convergence of multiple sequences Arch. Math. (Basel), 81 (1) (2003), 82–89.
14. F. Móricz: Statistical convergence of Walsh-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 20 (2) (2004), 165–168.
15. F. Móricz: Statistical convergence of sequences and series of complex numbers with applications in Fourier analysis and summability, Anal. Math., 39 (4) (2013), 271–285.
16. F. Móricz: Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series, Anal. Math., 43 (1) (2017), 103–116.
17. H. Oruc, G. M. Phillips: A generalization of the Bernstein polynomials, Proc. Edinburgh Math. Soc. (2), 42 (2) (1999), 403-413.
18. G. M. Phillips: A survey of results on the q-Bernstein polynomials, IMA J. Numer. Anal., 30 (1) (2010), 277–288.
19. G. M. Phillips: On generalized Bernstein polynomials. Numerical analysis, 263–269, World Sci. Publ., River Edge, NJ, (1996).
20. Webpage: https://www.mathcounterexamples.net/continuous-function-with-divergent-fourier-series
21. A. Zygmund: Trigonometric series, Vol. I and II. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002).