### Title page for ETD etd-07122012-095333

Type of Document Dissertation
Author Ho, Vivian Mankau
URN etd-07122012-095333
Title Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Ólafsson, Gestur Committee Chair
Davidson, Mark Committee Member
He, Hongyu Committee Member
Hoffman, Jerome Committee Member
Litherland, Richard Committee Member
Wu, Hsiao-Chun Dean's Representative
Keywords
• $chi$-spherical representations
• invariant differential operators
• Paley-Wiener Theorem
• compact symmetric spaces
• $chi$-spherical Fourier transforms
• hypergeometric functions
Date of Defense 2012-06-29
Availability unrestricted
Abstract
We generalize a Paley-Wiener theorem to homogeneous line bundles $L_\chi$ on a compact symmetric space U/K with $\chi$ a nontrivial character of K. The Fourier coefficients of a $\chi$-bi-coinvariant function f on U are defined by integration of f against the elementary spherical functions of type $\chi$ on U, depending on a spectral parameter $\mu$, which in turn parametrizes the $\chi$-spherical representations $\pi$ of U. The Paley-Wiener theorem characterizes f with sufficiently small support in terms of holomorphic extendability and exponential growth of their $\chi$-spherical Fourier transforms. We generalize Opdam's estimate for the hypergeometric functions in a bigger domain with the multiplicity parameters being not necessarily positive, which is crucial to the proof of Paley-Wiener theorem in our case.
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