ABSTRACT

Park, Robert Reeve, B.S., The University of California, Davis, 1985, M.S., The University of Idaho, 1990 Doctor of Philosophy, Spring Commencement, 1994, Louisiana State University Major: Mathematics A Paley-Wiener Theorem for All Two and Three-Step Nilpotent Lie Groups Dissertation directed by Professor Leonard Richardson

A Paley-Wiener Theorem for all connected, simply-connected two and threestep nilpotent Lie groups is proved. If φ ϵ L_{c}^{∞}(G), where G is a connected, simplyconnected two or three-step nilpotent Lie group such that the operator-valued Fourier transform φ̂(π) = 0 for all π in E, a subset of Ĝ of positive Plancherel measure, then it is shown that φ = 0 a. e. on G. The proof uses representationtheoretic methods from Kirillov theory for nilpotent Lie groups, and uses an integral formula for the operator-valued Fourier transform φ̂(π). It is also shown by example that the condition that G be simply-connected is necessary.