Title page for ETD etd-07052011-113126


Type of Document Dissertation
Author Dann, Susanna
Author's Email Address sdann@math.lsu.edu
URN etd-07052011-113126
Title Paley-Wiener 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
  • Paley-Wiener theorem
  • Projective limits
Date of Defense 2011-07-01
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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