Abstract
It is established that the formulas determining the jump of a periodic function from the derivatives of the partial sums of its Fourier series and valid for functions of harmonic bounded variation (the HBV class) possibly will not hold for functions of Φ-bounded variation (in the sense of Schramm) if this class is wider than the HBV class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. I. Golubov, “Determination of the jump of a function of bounded p-variation by its Fourier series,” Mat. Zametki 12 (1), 19–28 (1972) [Math. Notes 12 (1–2), 19–28 (1972)].
N. Wiener, “The quadratic variation a function and its Fourier coefficients,” J. Math. Phys. 3 (2), 72–94 (1924).
L. Fejér, “Über die Bestimmung des Sprunges der Funktion aus Fourier-reihe,” J. Reine Angew.Math. 142 (1), 165–188 (1912).
P. Czillag, “Über die Fourierkonstanten einer Funktion von Beschränkter Schwankung,” Mat. Phys. Lapok 27, 301–308 (1918).
A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959; Mir, Moscow, 1965), Vol. 1.
M. Avdispahić, “On the determination of the jump of a function by its Fourier series,” Acta Math. Hungar. 48 (3), 267–271 (1986).
G. Kvernadze, “Determination of the jumps of a bounded function by its Fourier series,” J. Approx. Theory 92 (2), 167–190 (1998).
D. Waterman, “On convergence of Fourier series of the functions of generalized bounded variation,” Studia Math. 44 (2), 107–117 (1972).
M. Schramm, “Functions of Φ-bounded variation and Riemann–Stieltjes integration,” Trans. Amer. Math. Soc. 287 (1), 49–63 (1985).
L. C. Young, “Sur une généralisation de la notion de variation de puissance p-ième bornée au sens de M.Wiener, et sur la convergence des séries de Fourier,” C. R. Acad. Sci., Paris 204 (7), 470–472 (1937).
L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nevskii Dialekt, St. Petersburg., 2004) [in Russian].
A. A. Kel’zon, “On the determination of the jump of a function of Φ-bounded variation from the derivatives of the partial sums of the Fourier series,” in Problems of Current Interest in Function Theory and Their Applications, Proceedings of the 17th Saratov International Winter Workshop (Izd. “Nauchnaya Kniga,” Saratov, 2014), pp. 116–118 [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. A. Kel’zon, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 1, pp. 35–41.
Rights and permissions
About this article
Cite this article
Kel’zon, A.A. Determination of the jump of a function of generalized bounded variation from the derivatives of the partial sums of its Fourier series. Math Notes 99, 46–51 (2016). https://doi.org/10.1134/S0001434616010053
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616010053