A random $L^{1}$ function with divergent Walsh series
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- by Benjamin B. Wells
- Proc. Amer. Math. Soc. 24 (1970), 794-796
- DOI: https://doi.org/10.1090/S0002-9939-1970-0261255-0
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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
- 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 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 10.1090/S0002-9947-1949-0032833-2
- Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Company Raytheon Education Company, 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 10.2307/1970308
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- 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