Sets of uniqueness in compact, 0-dimensional metric groups

Grubb, D. J.

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

Sets of uniqueness in compact, $0$-dimensional metric groups
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Trans. Amer. Math. Soc. 301 (1987), 239-249 Request permission

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.

References

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, DOI 10.1090/S0002-9939-1982-0637169-9
K. Yoneda, On generalized uniqueness theorems for Walsh series, Acta Math. Hungar. 43 (1984), no. 3-4, 209–217. MR 733855, DOI 10.1007/BF01958020
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
Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
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-Berlin, 1970. MR 0262773
N. J. Fine, Fourier-Stieltjes series of Walsh functions, Trans. Amer. Math. Soc. 86 (1957), 246–255. MR 91371, DOI 10.1090/S0002-9947-1957-0091371-6

Trigonometric series

Richard B. Crittenden and Victor L. Shapiro, Sets of uniqueness on the group $2^{\omega }$, Ann. of Math. (2) 81 (1965), 550–564. MR 179535, DOI 10.2307/1970401
William R. Wade, A uniqueness theorem for Haar and Walsh series, Trans. Amer. Math. Soc. 141 (1969), 187–194. MR 243265, DOI 10.1090/S0002-9947-1969-0243265-9
William R. Wade, Growth conditions and uniqueness for Walsh series, Michigan Math. J. 24 (1977), no. 2, 153–155. MR 487247
Kaoru Yoneda, Summing generalized closed $U$-sets for Walsh series, Proc. Amer. Math. Soc. 94 (1985), no. 1, 110–114. MR 781066, DOI 10.1090/S0002-9939-1985-0781066-8
William R. Wade, Summing closed $U$-sets for Walsh series, Proc. Amer. Math. Soc. 29 (1971), 123–125. MR 279522, DOI 10.1090/S0002-9939-1971-0279522-4