such vectors in . Moreover, if the group is unimodular, i.e., if the left and the right invariant measures coincide, then the existence of one admissible vector implies that every vector in is admissible. Given a square integrable representation and an admissible vector , let us define the vectors These vectors are the analogues of the canonical coherent states, written there in terms of the representation of the Heisenberg group (however, see the section on Gilmore-Perelomov CS, below). Next, it can be shown that the resolution of the identity holds on . Thus, the vectors constitute a family of generalized coherent states. The functions for all vectors in are square integrable with respect to the measure and the set of such functions, which in fact are continuous in the topology of , forms a closed subspace of . Furthermore, the mapping is a linear isometry between and and under this isometry the representation gets mapped to a subrepresentation of the left regular representation of on . An example: wavelets A typical example of the above construction is provided by the affine group of the line, . This is the group of all 2×2 matrices of the type, and being real numbers with . We shall also write , with the action on given by . This group is non-unimodular, with the left invariant measure being given by (the right invariant measure being ). The affine group has a unitary irreducible representation on the Hilbert space . Vectors in are measurable functions of the real variable and the (unitary) operators of this representation act on them as If is a function in such that its Fourier transform satisfies the (admissibility) condition then it can be shown to be an admissible vector, i.e., Thus, following the general construction outlined above, the vectors define a family of generalized coherent states and one has the resolution of the identity on . In the signal analysis literature, a vector satisfying the admissibility condition above is called a mother wavelet and the generalized coherent states are called wavelets. Signals are then identified with vectors in and the function is called the continuous wavelet transform of the signal . This concept can be extended to two dimensions, the group being replaced by the so-called similitude group of the plane, which consists of plane translations, rotations and global dilations. The resulting 2D wavelets, and some generalizations of them,