Sets of uniqueness in compact, 0-dimensional metric groups

Grubb, D. J.

doi:10.1090/S0002-9947-1987-0879571-5

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Sets of uniqueness in compact, 0-dimensional metric groups

Author: D. J. Grubb
Journal: Trans. Amer. Math. Soc. 301 (1987), 239-249
MSC: Primary 42C10; Secondary 43A46
MathSciNet review: 879571
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Abstract: An investigation is made of sets of uniqueness in a compact 0-dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary's theorem on countable unions of closed sets of uniqueness.

[1] William R. Wade and Kaoru Yoneda, Uniqueness and quasimeasures on the group of integers of a 𝑝-series field, Proc. Amer. Math. Soc. 84 (1982), no. 2, 202–206. MR 637169 (83c:43010), http://dx.doi.org/10.1090/S0002-9939-1982-0637169-9
[2] K. Yoneda, On generalized uniqueness theorems for Walsh series, Acta Math. Hungar. 43 (1984), no. 3-4, 209–217. MR 733855 (85k:42056), http://dx.doi.org/10.1007/BF01958020
[3] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. MR 0032833 (11,352b), http://dx.doi.org/10.1090/S0002-9947-1949-0032833-2
[4] Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121 (51 #3363)
[5] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York, 1970. MR 0262773 (41 #7378)
[6] N. J. Fine, Fourier-Stieltjes series of Walsh functions, Trans. Amer. Math. Soc. 86 (1957), 246–255. MR 0091371 (19,957a), http://dx.doi.org/10.1090/S0002-9947-1957-0091371-6
[7] A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1979.
[8] Richard B. Crittenden and Victor L. Shapiro, Sets of uniqueness on the group 2^{𝜔}, Ann. of Math. (2) 81 (1965), 550–564. MR 0179535 (31 #3783)
[9] William R. Wade, A uniqueness theorem for Haar and Walsh series, Trans. Amer. Math. Soc. 141 (1969), 187–194. MR 0243265 (39 #4587), http://dx.doi.org/10.1090/S0002-9947-1969-0243265-9
[10] William R. Wade, Growth conditions and uniqueness for Walsh series, Michigan Math. J. 24 (1977), no. 2, 153–155. MR 0487247 (58 #6900)
[11] Kaoru Yoneda, Summing generalized closed 𝑈-sets for Walsh series, Proc. Amer. Math. Soc. 94 (1985), no. 1, 110–114. MR 781066 (86g:42045), http://dx.doi.org/10.1090/S0002-9939-1985-0781066-8
[12] William R. Wade, Summing closed 𝑈-sets for Walsh series, Proc. Amer. Math. Soc. 29 (1971), 123–125. MR 0279522 (43 #5244), http://dx.doi.org/10.1090/S0002-9939-1971-0279522-4

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