Sets of uniqueness for a certain class $\mathcal {M}_\varepsilon$ on the dyadic group
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- by Kaoru Yoneda PDF
- Proc. Amer. Math. Soc. 89 (1983), 279-284 Request permission
Abstract:
For each sequence $\varepsilon = \left \{ {{\varepsilon _n}} \right \}$ of real numbers which satisfies $\lim {\inf _{n \to \infty }}{\varepsilon _{{2^{n + 1}}}}/{\varepsilon _{{2^n}}} > 0$ and ${\varepsilon _n} \downarrow 0$ as $n \to \infty$, let ${\mathfrak {M}_\varepsilon }$ denote the set of all Walsh series $\mu \sim \sum \nolimits _{k = 0}^\infty {\hat \mu (k){w_k}(x)}$ such that $\sum \nolimits _{k = 0}^\infty {{\varepsilon _k}{{\left | {\hat \mu (k)} \right |}^2} < \infty }$. We give a necessary and sufficient condition for a subset of the dyadic group to be a set of uniqueness for ${\mathfrak {M}_\varepsilon }$.References
- 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
- Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
- William R. Wade, Uniqueness and $\alpha$-capacity on the group $2^{\omega }$, Trans. Amer. Math. Soc. 208 (1975), 309–315. MR 380255, DOI 10.1090/S0002-9947-1975-0380255-2
- 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
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 279-284
- MSC: Primary 42C25; Secondary 42C10, 43A70
- DOI: https://doi.org/10.1090/S0002-9939-1983-0712637-0
- MathSciNet review: 712637