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One of the important questions related to any integral transform on amanifold M or on a homogeneous space G/K is the description of the image of agiven space of functions. If M=G/K, where (G,K) is a Gelfand pair, then theharmonic analysis is closely related to the representations of G and the directintegral decomposition of L^2(M) into irreducible representations. We give ashort overview of the Fourier transform on such spaces and then ask if one candescribe the image of the space of smooth compactly supported functions interms of the spectral parameter, i.e., the parameterization of the set ofirreducible representations in the support of the Plancherel measure forL^2(M). We then discuss the Euclidean motion group, semisimple symmetricspaces, and some limits of those spaces.

Paley-Wiener Theorems with respect to the spectral parameter