Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T13:41:32.597Z Has data issue: false hasContentIssue false
  • English
  • Français

On a recent theorem of Carleson. II

Published online by Cambridge University Press:  24 October 2008

B. S. Yadav
B. S. Yadav
Affiliation:
Sardar Patel University, Vallabh Vidyanagar, India
Get access Rights & Permissions [Opens in a new window]

Extract

The long open question regarding the almost everywhere convergence of the Fourier series of an L2-function having been recently settled by Carleson (1), an interesting problem still awaits solution. It is to investigate whether the Fourier series of a function of class Lp (1 < p < 2), converges (or diverges) almost everywhere; for it is already known that the Fourier series of an integrable function may diverge almost everywhere. See Kolmogorov (3) and Marcinkiewicz (4) or ((8), 1, p. 305), see also Kahane(2). The object of this paper is simply to show that a suitable restriction imposed on the modulus of smoothness of an integrable function guarantees the almost everywhere convergence of its Fourier series. This will establish, in a more general form, the author's conjecture made in (5). Our proof is based on Carleson's theorem according to which the Fourier series of an L2-function converges almost everywhere.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 1057 - 1059
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Carleson, L.On convergence and growth of partial sums of Fourier series. Acta Math. 116 (1966), 135–157.CrossRefGoogle Scholar
(2)Kahane, J. P.Sommes partielles des séries de Fourier. Séminaire N. Bourbaki 18 (June, 1966).Google Scholar
(3)Kolmogorov, A. N.Une série de Fourier-Lebesgue divergente presque partout. Fund. Math. 4 (1923), 324–329.CrossRefGoogle Scholar
(4)Marcinkiewicz, J.Sur les séries de Fourier. Fund. Math. 27 (1936), 38–69.CrossRefGoogle Scholar
(5)Yadav, B. S.On the almost everywhere convergence of Fourier series. Proc. Cambridge Philos. Soc. 63 (1967), 703–706.CrossRefGoogle Scholar
(6)Žuk, V. V.On the absolute convergence of Fourier series. Dokl. Akad. Nauk SSSR 160 (1965), 519–522 = Soviet Math. Dokl. 6 (1965), 120–123.Google Scholar
(7)Žuk, V. V.On a modification of the concept of the modulus of smoothness and its uses in estimating Fourier coefficients. Dokl. Akad. Nauk SSSR 160 (1965), 758–761 = Soviet Math. Dokl. 6 (1965), 183–187.Google Scholar
(8)Zygmund, A.Trigonometric Series, vols. I and II (Cambridge, 1959).Google Scholar