## LSU Doctoral Dissertations

#### Identifier

etd-07122012-095333

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### 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.

2012

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Ólafsson, Gestur

COinS