Type of Document 
Dissertation 
Author 
Dann, Susanna

Author's Email Address 
sdann@math.lsu.edu 
URN 
etd07052011113126 
Title 
PaleyWiener Theorems with Respect to the Spectral Parameter 
Degree 
Doctor of Philosophy (Ph.D.) 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Olafsson, Gestur 
Committee Chair 
Davidson, Mark 
Committee Member 
Litherland, Richard A. 
Committee Member 
Adkins, William 
Committee Member 
Hah, Doo Young 
Dean's Representative 

Keywords 
 Gelfand pairs
 Vector valued Fourier transform
 Euclidean motion group
 PaleyWiener theorem
 Projective limits

Date of Defense 
20110701 
Availability 
unrestricted 
Abstract
One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then harmonic analysis on M is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations of G. R^n can be realized as the quotient R^n=E(n)/SO(n), where E(n) is the orientation preserving Euclidean motion group. The pair (E(n), SO(n)) is a Gelfand pair. Hence this realization of R^n comes with its own natural Fourier transform derived from the representation theory of E(n). The representations of E(n) that are in the support of the Plancherel measure for L^2(R^n) are parameterized by R^+. We describe the image of smooth compactly supported functions under the Fourier transform with respect to the spectral parameter. Then we discuss an extension of our description to projective limits of corresponding function spaces.

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