Abstract
We construct an example of a function from the class \(H_1^{\omega ^ * } \), where \(\omega ^ * (t) = \sqrt {\log \log (t^{ - 1} )/\log (t^{ - 1} )} \), \(0 < t \leqslant t_0 \), whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes \(H_1^\omega \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
N.P. Korneichuk,Extremum Problems of Approximation Theory [in Russian],Nauka, Moscow,1976.
A. Zygmund,Trigonometric Series vol.2,Cambridge Univ.Press, Cambridge,1960.Russian translation:Mir, Moscow, 1965.
A. Kolmogoro.,“Une série de Fourier–Lebesgue divergente presque partout,”Fund. Math.,4 (1923), 324–328.
A. Kolmogoro.,“Une série de Fourier–Lebesgue divergente partout,”C. R. Acad. Sci. Paris 183 (1926),1327–1329.
V.I. Prokhorenko,“On divergent Fourier series,”Mat. Sb. [Math. USSR-Sb.],75(117) (1968),no.2, 185–198.
S.V. Konyagin,“On divergence everywhere of trigonometric Fourier series,”Mat. Sb. [Russian Acad. Sci. Sb. Math.],191 (2000),no.1,103–126.
Y.M. Chen,“On Kolmogoroff's divergent Fourier series,”Archiv der Mathematik 14 (1963),no.2, 116–119.
Y.M. Chen,“An almost ewerywhere divergent Fourier series of class L(log+ log+ L)1-å,”J. London Math. Soc.,44 (1969),643–654.
K. Tandori,“Ein Divergenzsats f#x00FC;r Fourierreihen,”Acta Sci. Math.,30 (1969), 43–48.
T.W. Körner,“Everywhere divergent Fourier series,”Colloq. Math.,45 (1981),no.1, 103–118.
G.H. Hardy,“On the summability of Fourier series,”Proc. London Math. Soc.,12 (1913), 365–372.
P.L. Ul'yanov,“On series with respect to the Haar system,”Mat. Sb. [Math. USSR-Sb.],63 (105) (1964),no.3,356–391.
P.L. Ul'yanov, “Embedding of some classes of functions H w p ” Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],32 (1968),no.3, 649–686.
N.K. Bari,Trigonometric Series [in Russian],Gos.Izdat.Fiz.-Mat.Lit., Moscow,1961.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Antonov, N.Y. Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity. Mathematical Notes 76, 606–619 (2004). https://doi.org/10.1023/B:MATN.0000049660.29081.bc
Issue Date:
DOI: https://doi.org/10.1023/B:MATN.0000049660.29081.bc