Sets of uniqueness for Vilenkin series of bounded type

Author

Stanley O. Perrine

Date of Award

5-1997

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

William R. Wade

Committee Members

Kenneth Stephenson, J. Milton Bailey, Balram Rajput

Abstract

The main technique used in many of the proofs found herein is fundamental real analysis measure theory. Because every Vilenkin series has an associated quasi-measure, the group analysis can be performed on the interval (0,1), instead of on the group itself. Also, the orthogonality of the group of Vilenkin characters plays a vital role in these results.

In this dissertation, it is shown that any perfect set of Haar measure zero is a set of multiplicity for C. It is also shown that having α-capacity zero is a necessary and sufficient condition for a set to be a set of uniqueness for the collection T_α⁺. Chapters 5 and 6 generalize this last result: one result from Chapter 5 actually shows that the previous theorem holds true for Tα in place of T_α⁺; Chapter 6 introduces a generalized ε-capacity, for which it is shown that a set of ε-capacity zero is a set of uniqueness for U_ε⁺. Finally, it is shown that the small collection Β has a large number of sets of uniqueness, but that there does exist a set of multiplicity for Β which has Haar measure zero.

Recommended Citation

Perrine, Stanley O., "Sets of uniqueness for Vilenkin series of bounded type. " PhD diss., University of Tennessee, 1997.
https://trace.tennessee.edu/utk_graddiss/9581