A random $L^{1}$ function with divergent Walsh series
Author:
Benjamin B. Wells
Journal:
Proc. Amer. Math. Soc. 24 (1970), 794-796
MSC:
Primary 42.16
DOI:
https://doi.org/10.1090/S0002-9939-1970-0261255-0
MathSciNet review:
0261255
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Abstract | References | Similar Articles | Additional Information
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.
- 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 217510, DOI https://doi.org/10.4064/sm-28-3-363-388
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 32833, DOI https://doi.org/10.1090/S0002-9947-1949-0032833-2
- Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0254888
- E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 125392, DOI https://doi.org/10.2307/1970308
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Additional Information
Keywords:
Fourier-Walsh series,
random function,
almost surely divergent series
Article copyright:
© Copyright 1970
American Mathematical Society