Paley-Wiener theorems for the Theta-spherical transform: an overview

The aim of this article is to give an overview of several types of Paley-Wiener theorems occuring in harmonic analysis related to symmetric spaces. This will serve as a motivation for the introduction of the $\Theta$-spherical functions, the correspondint $\Theta$-spherical Fourier transform, and the Paley-Wiener theorem for this transform. Up to now such a theorem has only been proven in very special cases, and still, its formulation and proof are very technical. In this paper we do not go into details of the proofs, but present an overview which explains the different examples which have inspired and motivated the theory of $\Theta$-spherical functions.