Doctor of Philosophy (PhD)
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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Ho, Vivian Mankau, "Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces" (2012). LSU Doctoral Dissertations. 2195.