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Almost everywhere convergence of Vilenkin-Fourier series of $ H\sp 1$ functions


Author: Wo-Sang Young
Journal: Proc. Amer. Math. Soc. 108 (1990), 433-441
MSC: Primary 42C10
DOI: https://doi.org/10.1090/S0002-9939-1990-0998742-6
MathSciNet review: 998742
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Abstract: In [5] Ladhawala and Pankratz proved that if $ f$ is in dyadic $ {H^1}$, then any lacunary sequence of partial sums of the Walsh-Fourier series of $ f$ converges a.e. We generalize their theorem to Vilenkin-Fourier series. In obtaining this result, we prove a vector-valued inequality for the partial sums of Vilenkin-Fourier series.


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

  • [1] B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187-190. MR 0268966 (42:3863)
  • [2] S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Math. Hung. 45 (1985), 223-234. MR 779509 (86m:42032)
  • [3] A. M. Garsia, Martingale inequalities: Seminar notes on recent progress, Benjamin, Reading, Massachusetts, 1973. MR 0448538 (56:6844)
  • [4] J. A. Gosselin, Almost everywhere convergence of Vilenkin-Fourier series, Trans. Amer. Math. Soc. 185 (1973), 345-370. MR 0352883 (50:5369)
  • [5] N. R. Ladhawala and D. C. Pankratz, Almost everywhere convergence of Walsh Fourier series of $ {\mathcal{H}^1}$-functions, Studia Math. 59 (1976/77), 85-92. MR 0430663 (55:3668)
  • [6] P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984), 87-101 (1985). MR 823096 (87b:42032)
  • [7] N. Ja. Vilenkin, On a class of complete orthonormal systems, Amer. Math. Soc. Transl. (2) 28 (1963), 1-35. MR 0154042 (27:4001)
  • [8] W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311-320. MR 0394022 (52:14828)

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DOI: https://doi.org/10.1090/S0002-9939-1990-0998742-6
Article copyright: © Copyright 1990 American Mathematical Society

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