Coherent States and Wavelets, a Contemporary Panorama

Abstract

In a first part, we review the general theory of coherent states (CS). Starting from the canonical CS introduced by Schrödinger in 1926 and rediscovered by Glauber, Klauder and Sudarshan in the 1960s, we proceed to the derivation of general CS from square integrable group representations and some of their generalizations: Perelomov CS, general square integrable covariant CS, nonlinear CS, Gazeau–Klauder CS, with a hint to their application in quantization.

Next, we turn to signal processing and note that two of the most familiar tools, namely, the Gabor transform and the wavelet transform, are special cases of CS, associated to the Weyl–Heisenberg group (which yields the canonical CS) and the affine group of the line, respectively. Then we review the properties of the wavelet transform, both in its continuous and its discrete versions, in one or two dimensions, emphasizing mostly the mathematical properties. We also consider its extension to higher dimensions, to more general manifolds (sphere, hyperboloid,. . . ) and to the space-time context, for the analysis of moving objects.

Mathematics Subject Classification (2010). 22D10, 42C40, 62H35, 68U10, 81R30