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Perfect Sets of Uniqueness on the Group 2ω

Published online by Cambridge University Press:  20 November 2018

Kaoru Yoneda
Kaoru Yoneda*
Affiliation:
University of Osaka Prefecture, Sakai, Osaka, Japan
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Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfies

A subset E of G which is not a set of uniqueness is called a set of multiplicity.

It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 3 , 01 June 1982 , pp. 759 - 764
Copyright
Copyright © Canadian Mathematical Society 1982

References

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