Sets of divergence on the group $2^{\omega }$
HTML articles powered by AMS MathViewer
- by David C. Harris and William R. Wade PDF
- Trans. Amer. Math. Soc. 240 (1978), 385-392 Request permission
Abstract:
We show that there exist uncountable sets of divergence for $C({2^\omega })$. We also show that a necessary and sufficient condition that a set E be a set of divergence for ${L^p}({2^\omega }),1 < p < \infty$, is that E be of Haar measure zero.References
- P. Billard, Sur la convergence presque partout des sĂ©ries de Fourier-Walsh des fonctions de lâespace $L^{2}\,(0,\,1)$, Studia Math. 28 (1966/67), 363â388. MR 217510, DOI 10.4064/sm-28-3-363-388
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372â414. MR 32833, DOI 10.1090/S0002-9947-1949-0032833-2
- L. H. Harper, Capacities of sets and harmonic analysis on the group $2^{\omega }$, Trans. Amer. Math. Soc. 126 (1967), 303â315. MR 206627, DOI 10.1090/S0002-9947-1967-0206627-X
- Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
- V. I. Prohorenko, Divergent Fourier series with respect to Haarâs system, Izv. VysĆĄ. UÄebn. Zaved. Matematika 1(104) (1971), 62â68 (Russian). MR 0294983
- F. Schipp, Construction of a continuous function whose Walsh series diverges at a prescribed point, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 9 (1966), 103â108. MR 206612
- Ferenc Schipp, Ăber die Grössenordnung der Partialsummen der Entwicklung integrierbarer Funktionen nach $W$-Systemen, Acta Sci. Math. (Szeged) 28 (1967), 123â134 (German). MR 213810
- Per Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series, Ark. Mat. 7 (1969), 551â570. MR 241885, DOI 10.1007/BF02590894
- E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140â170. MR 125392, DOI 10.2307/1970308 W. R. Wade, Sets of divergence of the group ${2^\omega }$, Notices Amer. Math. Soc. 21 (1974), A-162. Abstract #711-42-1.
- William R. Wade, A uniqueness theorem for Haar and Walsh series, Trans. Amer. Math. Soc. 141 (1969), 187â194. MR 243265, DOI 10.1090/S0002-9947-1969-0243265-9
- J. L. Walsh, A Closed Set of Normal Orthogonal Functions, Amer. J. Math. 45 (1923), no. 1, 5â24. MR 1506485, DOI 10.2307/2387224
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 240 (1978), 385-392
- MSC: Primary 42A56
- DOI: https://doi.org/10.1090/S0002-9947-1978-0487242-7
- MathSciNet review: 0487242