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Paley-Wiener type theorems describe the image of a given space of functions,often compactly supported functions, under an integral transform, usually aFourier transform on a group or homogeneous space. Several authors have studiedPaley-Wiener type theorems for Euclidean spaces, Riemannian symmetric spaces ofcompact or non-compact type as well as affine Riemannian symmetric spaces. Inthis article we prove a Paley-Wiener theorem for homogeneous line bundles overa compact symmetric space $U/K$. The Paley-Wiener theorem characterizes f withsufficiently small support in terms of holomorphic extendability andexponential growth of their Fourier transforms. An important tool is ageneralization of Opdam's estimate for the hypergeometric functions formultiplicity functions that are not necessarily positive. The domain where thisestimate is valid is also bigger. This is done in an appendix.

Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces and New Estimates for the Heckman-Opdam Hypergeometric Functions