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A random $ L\sp{1}$ function with divergent Walsh series

Author: Benjamin B. Wells
Journal: Proc. Amer. Math. Soc. 24 (1970), 794-796
MSC: Primary 42.16
MathSciNet review: 0261255
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Abstract: The purpose of this paper is to point out that the techniques of J. P. Kahane to arrive at almost everywhere divergent Fourier series may be carried over to the Fourier-Walsh system. In particular we construct a random $ {L^1}$ function whose Fourier-Walsh series almost surely (a. s.) diverges almost everywhere.

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

  • [1] P. Billard, Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l'espace $ {L^2}(0,\;1)$, Studia Math. 28 (1966/67), 363-388. MR 36 #599. MR 0217510 (36:599)
  • [2] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. MR 11, 352. MR 0032833 (11:352b)
  • [3] J. P. Kahane, Some random series of functions, Heath, Lexington, Mass., 1968. MR 0254888 (40:8095)
  • [4] E. M. Stein, On limits of sequences of operators, Ann. of Math. (2) 74 (1961), 140-170. MR 23 #A2695. MR 0125392 (23:A2695)

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Keywords: Fourier-Walsh series, random function, almost surely divergent series
Article copyright: © Copyright 1970 American Mathematical Society

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