Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the absolute convergence of lacunary Fourier series


Authors: N. V. Patel and V. M. Shah
Journal: Proc. Amer. Math. Soc. 93 (1985), 433-439
MSC: Primary 42A55; Secondary 42A28
DOI: https://doi.org/10.1090/S0002-9939-1985-0773997-X
MathSciNet review: 773997
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: P. B. Kennedy [3] studied lacunary Fourier series whose generating functions are of bounded variation on a subinterval $ I$ of $ [ - \pi ,\pi ]$ and satisfy a Lispschitz condition of order $ \alpha $ on $ I$. We show that the conclusion of one of his theorems on the absolute convergence of Fourier series remains valid when the function is merely of bounded $ r$th variation in $ I$ and belongs to a class $ \operatorname{Lip}(\alpha ,p)$ in $ I$. Our results also generalize three theorems of S. M. Mazhar [4].


References [Enhancements On Off] (What's this?)

  • [1] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), 565-606. MR 1544927
  • [2] -, Notes on the theory of series. IX: On the absolute convergence of Fourier series, J. London Math. Soc. 3 (1928), 250-253.
  • [3] P. B. Kennedy, Fourier series with gaps, Quart. J. Math. Oxford Ser. (2) 7 (1956), 224-230. MR 0098272 (20:4733)
  • [4] S. M. Mazhar, On the absolute convergence of a Fourier series with gaps, Proc. Nat. Acad. Sci. India Sect. A 26 (1960), 104-109. MR 0126122 (23:A3418)
  • [5] M. E. Noble, Coefficient properties of Fourier series with a gap condition, Math. Ann. 128 (1954), 55-62. MR 0063469 (16:126a)
  • [6] R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., Vol. 19, Amer. Math. Soc., Providence, R. I., 1934. MR 1451142 (98a:01023)
  • [7] A. Zygmund, Trigonometrical series, PWN, Warsaw, 1935.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A55, 42A28

Retrieve articles in all journals with MSC: 42A55, 42A28


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0773997-X
Keywords: Lacunary Fourier series, absolute convergence, Lipschitz condition, bounded $ r$th variation
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society