## Sets of uniqueness in compact, $0$-dimensional metric groups

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- by D. J. Grubb PDF
- 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. Baryβs 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.Β 2, 202β206. 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.Β 3-4, 209β217. MR**733855**, DOI 10.1007/BF01958020 - N. J. Fine,
*On the Walsh functions*, Trans. Amer. Math. Soc.**65**(1949), 372β414. 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), 246β255. MR**91371**, DOI 10.1090/S0002-9947-1957-0091371-6
A. Zygmund, - 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 10.2307/1970401 - William R. Wade,
*A uniqueness theorem for Haar and Walsh series*, Trans. Amer. Math. Soc.**141**(1969), 187β194. 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.Β 2, 153β155. MR**487247** - Kaoru Yoneda,
*Summing generalized closed $U$-sets for Walsh series*, Proc. Amer. Math. Soc.**94**(1985), no.Β 1, 110β114. 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), 123β125. MR**279522**, DOI 10.1090/S0002-9939-1971-0279522-4

*Trigonometric series*, Cambridge Univ. Press, Cambridge, 1979.

## Additional Information

- © Copyright 1987 American Mathematical Society
- 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