Abstract
The class of generalized bounded variation is considered. For functions from this class, the deviation of the generalized Cesáro means of negative order in the norm of Lr, (1 ≤ r ≤ ∞) is estimated.
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References
C. Jordan, “Sur la série de Fourier,” C. R. Acad. Sci. Paris 92, 228–230 (1881).
N. Wiener, “The quadratic variation of a function and its Fourier coefficients,” Massachusetts J. Math. 3 (2), 72–94 (1924).
L. Young, “Sur une généralization de la notion de variation de puissance p-ième borneé au sens de N. Wiener, et sur la convergence des seriés de Fourier,” C. R. Acad. Sci. Paris 204 (1), 470–472 (1937).
D. Waterman, “On the summability of Fourier series of functions of A-bounded variation,” Studia Math. 44 (2), 107–117 (1972).
Z. Chanturia, “The modulus of variation of a function and its application in the theory of Fourier series,” Dokl. Akad. Nauk. SSSR 214 (1), 63–66 (1974).
T. Akhobadze, “Functions of generalized Wiener classes BV (p(n) ↑ ∞, φ) and their Fourier coefficients,” Georgian Math. J. 7 (3), 401–416 (2000).
H. Kita and K. Yoneda, “A generalization of bounded variation,” Acta Math. Hungar. 56 (3–4), 229–238 (1990).
H. Kita, “Convergence of Fourier series of a function of generalized Wiener’s class BV (p(n) ↑ p),” Acta Math. Hungar. 57 (3–4), 233–243 (1991).
T. Akhobadze, “A generalization of bounded variation,” Acta Math. Hungar. 97 (3), 223–256 (2002).
R. Siddiqi, “The order of Fourier coefficients of higher variation,” Proc. Japan Acad. 48 (7), 569–572 (1972).
A. Zigmund, Trigonometric Series (Cambridge University Press, 2002), Vols. I, II.
I. B. Kaplan, “Cesáro means of variable order,” Izv. Vyssh. Uchebn. Zaved. Mat. 18 (5), 62–73 (1960) [in Russian].
T. Akhobadze, “On the convergence of generalized Cesáro means of trigonometric Fourier series I,” Acta Math. Hungar. 115 (1–2), 59–78 (2007).
T. Akhobadze, “On a theorem of M. Satô,” Acta Math. Hungar. 130 (3), 286–308 (2011).
Sh. Tetunashvili, “On iterated summability of trigonometric Fourier series,” Proc. A. Razmadze Math. Inst. 139, 142–144 (2005).
Sh. Tetunashvili, “On the summability of Fourier trigonometric series of variable order,” Proc. A. Razmadze Math. Inst. 145, 130–131 (2007).
Sh. Tetunashvili, “On the summability method defined by matrix of functions,” Proc. A. Razmadze Math. Inst. 148, 141–145 (2008).
Sh. Tetunashvili, “On the summability method depending on a parameter,” Proc. A. Razmadze Math. Inst. 150, 150–152 (2009).
Sh. Tetunashvili, “On divergence of Fourier trigonometric series by some methods of summability with variable orders,” Proc. A. Razmadze Math. Inst. 155, 130–131 (2011).
Sh. Tetunashvili, “On divergence of Fourier series by some methods of summability,” Journal of Function Spaces and Applications 2012 Article ID 542607 (2010).
G. Fichtengolz, Course of Differential and Integral Calculus (Nauka, 1970), Vol II [in Russian].
Acknowledgments
The authors are very grateful to the referee for the careful reading of the paper and helpful comments and remarks, which have allowed the authors to improve the quality of the paper.
Funding
This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) under grant no. FR-18-1599.
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Akhobadze, T.I., Zviadadze, S.V. On the Generalized Cesáro Summability of Trigonometric Fourier Series. Math Notes 107, 898–906 (2020). https://doi.org/10.1134/S000143462005020X
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DOI: https://doi.org/10.1134/S000143462005020X