#### Date of Award

1994

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Leonard Richardson

#### Abstract

A Paley-Wiener Theorem for all connected, simply-connected two and three-step nilpotent Lie groups is proved. If f $\epsilon \ L\sbsp{c}{\infty}({G}),$ where G is a connected, simply-connected two or three-step nilpotent Lie group such that the operator-valued Fourier transform $\\varphi(\pi)$ = 0 for all $\pi$ in E, a subset of G of positive Plancherel measure, then it is shown that f = 0 a. e. on G. The proof uses representation-theoretic methods from Kirillov theory for nilpotent Lie groups, and uses an integral formula for the operator-valued Fourier transform $\\varphi(\pi)$. It is also shown by example that the condition that G be simply-connected is necessary.

#### Recommended Citation

Park, Robert Reeve, "A Paley-Wiener Theorem for All Two and Three-Step Nilpotent Lie Groups." (1994). *LSU Historical Dissertations and Theses*. 5749.

https://digitalcommons.lsu.edu/gradschool_disstheses/5749

#### Pages

69