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
DOI:
https://doi.org/10.1090/S0002-9947-1987-0879571-5
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. Baryβs theorem on countable unions of closed sets of uniqueness.
- 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, 202β206. MR 637169, DOI https://doi.org/10.1090/S0002-9939-1982-0637169-9
- K. Yoneda, On generalized uniqueness theorems for Walsh series, Acta Math. Hungar. 43 (1984), no. 3-4, 209β217. MR 733855, DOI https://doi.org/10.1007/BF01958020
- N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372β414. MR 32833, DOI https://doi.org/10.1090/S0002-9947-1949-0032833-2
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. 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), 246β255. MR 91371, DOI https://doi.org/10.1090/S0002-9947-1957-0091371-6 A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1979.
- Richard B. Crittenden and Victor L. Shapiro, Sets of uniqueness on the group $2^{\omega }$, Ann. of Math. (2) 81 (1965), 550β564. MR 179535, DOI https://doi.org/10.2307/1970401
- William R. Wade, A uniqueness theorem for Haar and Walsh series, Trans. Amer. Math. Soc. 141 (1969), 187β194. MR 243265, DOI https://doi.org/10.1090/S0002-9947-1969-0243265-9
- William R. Wade, Growth conditions and uniqueness for Walsh series, Michigan Math. J. 24 (1977), no. 2, 153β155. MR 487247
- Kaoru Yoneda, Summing generalized closed $U$-sets for Walsh series, Proc. Amer. Math. Soc. 94 (1985), no. 1, 110β114. MR 781066, DOI https://doi.org/10.1090/S0002-9939-1985-0781066-8
- William R. Wade, Summing closed $U$-sets for Walsh series, Proc. Amer. Math. Soc. 29 (1971), 123β125. MR 279522, DOI https://doi.org/10.1090/S0002-9939-1971-0279522-4
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Additional Information
Keywords:
Vilenkin group,
set of uniqueness
Article copyright:
© Copyright 1987
American Mathematical Society