We consider R 3 as a homogeneous manifold for the action of the motion group given by rotations and translations. For an arbitrary τ ∈ SO(3), let E τ be the homogeneous vector bundle over R 3 associated with τ . An interesting problem consists in studying the set of bounded linear operators over the sections of E τ that are invariant under the action of SO(3) ⋉ R 3 . Such operators are in correspondence with the End(V τ )-valued, bi-τ -equivariant, integrable functions on R 3 and they form a commutative algebra with the convolution product. We develop the spherical analysis on that algebra, explicitly computing the τ -spherical functions. We first present a set of generators of the algebra of SO(3) ⋉ R 3 -invariant differential operators on E τ . We also give an explicit form for the τ -spherical Fourier transform, we deduce an inversion formula and we use it to give a characterization of End(V τ )-valued, bi-τ -equivariant, functions on R 3 .
In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space H. We find necessary and sufficient conditions on a bounded linear operator A ∈ B(H) and a set of vectors G ⊂ H, in order to obtain that {e tA g} g∈G,t∈[0,∞) is a semi-continuous frame for H. We study if it is possible to discretize the time variable t and still have a frame for H. We also relate the continuous iteration e tA on a set G to the discrete iteration (A ′ ) n on G ′ for an adequate operator A ′ and set G ′ ⊂ H.
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