Paley-Wiener theorems with respect to the spectral parameter

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then harmonic analysis on M is closely related to the representations of G and the direct integral decomposition of the space of square-integrable functions on M into irreducible representations of G. The n-dimensional Euclidean space can be realized as the quotient of the orientation preserving Euclidean motion group E(n) by the special orthogonal group SO(n). The pair (E(n), SO(n)) is a Gelfand pair. Hence this realization of the n-dimensional Euclidean space comes with its own natural Fourier transform derived from the representation theory of E(n). The representations of E(n) that are in the support of the Plancherel measure for the space of square-integrable functions on n-dimensional Euclidean space are parameterized by positive reals. We describe the image of smooth compactly supported functions under the Fourier transform with respect to the spectral parameter. Then we discuss an extension of our description to projective limits of corresponding function spaces.