Rate of convergence of certain Fourier series of functions of generalized bounded variation
https://doi.org/10.26907/0021-3446-2024-10-3-17
Abstract
In this paper, we discuss the rate of convergence of the rational Fourier series and conjugate rational Fourier series of functions of generalized bounded variation. In particular, well- known Wiener’s and Siddiqi’s theorems for functions of p-bounded variation are proved in more general complete rational orthogonal system. Also some results are obtained for a class of functions wider than the class of functions of bounded variation and of {n\alpha }-bounded variation.
About the Authors
R. K. Bera
The Maharaja Sayajirao University of Baroda
India
India
Rameshbhai Karshanbhai Bera
Vadodara (Gujarat), 390 002
B. L. Ghodadra
The Maharaja Sayajirao University of Baroda
India
India
Bhikha Lila Ghodadra
Vadodara (Gujarat), 390 002
References
1. Jordan C. Sur la s´eries de Fourier, C. R. Acad. Sci. Paris. 92, 228–230 (1881).
2. Zygmund A. Trigonometric series, V. I (Cambridge Univ. Press, Cambridge, 1959).
3. Young W.H. Konvergenzbedingungen f¨ur die verwandte Reihe einer Fourierschen Reihe, Munch. Ber. (41), 361–371 (1911).
4. Tan L., Qian T. On convergence of rational Fourier series of functions of bounded variations, Sci. Sin. Math. 43 (6), 541–550 (2013).
5. Bojani´c R. An estimate of the rate of convergence for Fourier series of functions of bounded variation, Publ. Inst. Math. (Beograd) (N.S.) 26 (40), 57–60 (1979).
6. Bojani´c R., Waterman D. On the rate of convergence of Fourier series of functions of generalized bounded variation, Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka. 22, 5–11 (1983).
7. Khachar H.J., Vyas R.G. Rate of convergence for rational and conjugate rational Fourier series of functions of generalized bounded variation, Acta Comm. Univ. Tartuensis Math. 26 (2), 233–241 (2022).
8. Mazhar S.M., Al-Budaiwi A. An estimate of the rate of convergence of the conjugate Fourier series of functions of bounded variation, Acta Math. Hungarica 49, 377–380 (1987).
9. Wiener N. The Quadratic Variation of a Function and its Fourier Coefficients, Mass. J. Math. (3), 72–94 (1924).
10. Siddiqi R.N. On convergence of the conjugate Fourier series of a function of Wiener’s class, Bull. Australian Math. Soc. 39 (3), 335–338 (1989).
11. Bultheel A., Gonz`alez-Vera P., Hendriksen E., Njastad O. Orthogonal rational functions (Cambridge Univ. Press, 1999).
12. Ninness B., Gustafsson F. A unifying construction of orthonormal bases for system identification, IEEE Transactions on Automatic Control 42 (4), 515–521 (1997).
13. Vyas R.G. Properties of functions of generalized bounded variation, Math. Anal., Approx. Theory and Their Appl., 715–741 (2016).
14. Hormozi M., Ledari A., Prus-Wisniowski F. On p-Lambda -bounded variation, Bull. Iran. Math. Soc. 37 (4), 35–49
15. Bari N.K. A treatise on trigonometric series, V. I (Newyork: Pergamon, 1964).
16. Garnett J.B. Bounded Analytic Functions (Academic Press, 1981).
Review
For citations:
Bera R.K., Ghodadra B.L. Rate of convergence of certain Fourier series of functions of generalized bounded variation. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2024;(10):3-17. (In Russ.) https://doi.org/10.26907/0021-3446-2024-10-3-17
Views:
110
Email this article 