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Generalized Cesàro summability of Fourier series and its applications

Year 2021, , 135 - 144, 01.06.2021
https://doi.org/10.33205/cma.838606

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.

References

  • H. Aktuglu, S. Bekar: q-Cesàro matrix and q-statistical convergence, J. Comput. Appl. Math., 235 (16) (2011), 4717–4723.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, (1994).
  • G. A. Anastassiou, O. Duman: Towards intelligent modeling: statistical approximation theory, Intelligent Systems Reference Library, 14. Springer-Verlag, Berlin, (2011).
  • J. Bustoz, L. F. Gordillo: q-Hausdorff summability, J. Comput. Anal. Appl., 7 (1) (2005), 35–48.
  • J. S. Connor: The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1–2) (1988), 47–63.
  • O. Duman: Statistical approximation for periodic functions, Demonstratio Math., 36 (4) (2003), 873–878.
  • O. Duman, M. K. Khan and C. Orhan: A-statistical convergence of approximating operators, Math. Inequal. Appl., 6 (4) (2003), 689–699.
  • H. Fast: Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • J. A. Frid: On statistical convergence Analysis, 5 (4) (1985), 301–313.
  • A. D. Gadjiev, C. Orhan: Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (1) (2002), 129–138.
  • V. Kac, P. Cheung: Quantum calculus, Universitext. Springer-Verlag, New York, (2002).
  • 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).
  • F. Móricz: Statistical convergence of multiple sequences Arch. Math. (Basel), 81 (1) (2003), 82–89.
  • F. Móricz: Statistical convergence of Walsh-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 20 (2) (2004), 165–168.
  • 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.
  • F. Móricz: Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series, Anal. Math., 43 (1) (2017), 103–116.
  • H. Oruc, G. M. Phillips: A generalization of the Bernstein polynomials, Proc. Edinburgh Math. Soc. (2), 42 (2) (1999), 403-413.
  • G. M. Phillips: A survey of results on the q-Bernstein polynomials, IMA J. Numer. Anal., 30 (1) (2010), 277–288.
  • G. M. Phillips: On generalized Bernstein polynomials. Numerical analysis, 263–269, World Sci. Publ., River Edge, NJ, (1996).
  • Webpage: https://www.mathcounterexamples.net/continuous-function-with-divergent-fourier-series
  • A. Zygmund: Trigonometric series, Vol. I and II. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002).
Year 2021, , 135 - 144, 01.06.2021
https://doi.org/10.33205/cma.838606

Abstract

References

  • H. Aktuglu, S. Bekar: q-Cesàro matrix and q-statistical convergence, J. Comput. Appl. Math., 235 (16) (2011), 4717–4723.
  • F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17. Walter de Gruyter & Co., Berlin, (1994).
  • G. A. Anastassiou, O. Duman: Towards intelligent modeling: statistical approximation theory, Intelligent Systems Reference Library, 14. Springer-Verlag, Berlin, (2011).
  • J. Bustoz, L. F. Gordillo: q-Hausdorff summability, J. Comput. Anal. Appl., 7 (1) (2005), 35–48.
  • J. S. Connor: The statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1–2) (1988), 47–63.
  • O. Duman: Statistical approximation for periodic functions, Demonstratio Math., 36 (4) (2003), 873–878.
  • O. Duman, M. K. Khan and C. Orhan: A-statistical convergence of approximating operators, Math. Inequal. Appl., 6 (4) (2003), 689–699.
  • H. Fast: Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • J. A. Frid: On statistical convergence Analysis, 5 (4) (1985), 301–313.
  • A. D. Gadjiev, C. Orhan: Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (1) (2002), 129–138.
  • V. Kac, P. Cheung: Quantum calculus, Universitext. Springer-Verlag, New York, (2002).
  • 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).
  • F. Móricz: Statistical convergence of multiple sequences Arch. Math. (Basel), 81 (1) (2003), 82–89.
  • F. Móricz: Statistical convergence of Walsh-Fourier series, Acta Math. Acad. Paedagog. Nyházi. (N.S.), 20 (2) (2004), 165–168.
  • 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.
  • F. Móricz: Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series, Anal. Math., 43 (1) (2017), 103–116.
  • H. Oruc, G. M. Phillips: A generalization of the Bernstein polynomials, Proc. Edinburgh Math. Soc. (2), 42 (2) (1999), 403-413.
  • G. M. Phillips: A survey of results on the q-Bernstein polynomials, IMA J. Numer. Anal., 30 (1) (2010), 277–288.
  • G. M. Phillips: On generalized Bernstein polynomials. Numerical analysis, 263–269, World Sci. Publ., River Edge, NJ, (1996).
  • Webpage: https://www.mathcounterexamples.net/continuous-function-with-divergent-fourier-series
  • A. Zygmund: Trigonometric series, Vol. I and II. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002).
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Oktay Duman 0000-0001-7779-6877

Publication Date June 1, 2021
Published in Issue Year 2021

Cite

APA Duman, O. (2021). Generalized Cesàro summability of Fourier series and its applications. Constructive Mathematical Analysis, 4(2), 135-144. https://doi.org/10.33205/cma.838606
AMA Duman O. Generalized Cesàro summability of Fourier series and its applications. CMA. June 2021;4(2):135-144. doi:10.33205/cma.838606
Chicago Duman, Oktay. “Generalized Cesàro Summability of Fourier Series and Its Applications”. Constructive Mathematical Analysis 4, no. 2 (June 2021): 135-44. https://doi.org/10.33205/cma.838606.
EndNote Duman O (June 1, 2021) Generalized Cesàro summability of Fourier series and its applications. Constructive Mathematical Analysis 4 2 135–144.
IEEE O. Duman, “Generalized Cesàro summability of Fourier series and its applications”, CMA, vol. 4, no. 2, pp. 135–144, 2021, doi: 10.33205/cma.838606.
ISNAD Duman, Oktay. “Generalized Cesàro Summability of Fourier Series and Its Applications”. Constructive Mathematical Analysis 4/2 (June 2021), 135-144. https://doi.org/10.33205/cma.838606.
JAMA Duman O. Generalized Cesàro summability of Fourier series and its applications. CMA. 2021;4:135–144.
MLA Duman, Oktay. “Generalized Cesàro Summability of Fourier Series and Its Applications”. Constructive Mathematical Analysis, vol. 4, no. 2, 2021, pp. 135-44, doi:10.33205/cma.838606.
Vancouver Duman O. Generalized Cesàro summability of Fourier series and its applications. CMA. 2021;4(2):135-44.