  The squared eigenfunctions of the spectral problem associated to the
Camassa-Holm equation represent a complete basis of functions, which helps to
describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a
Generalised Fourier transform. The main result of this work is the derivation
of the completeness relation for the squared solutions of the Camassa-Holm
spectral problem. We show that all the fundamental properties of the
Camassa-Holm equation such as the integrals of motion, the description of the
equations of the whole hierarchy and their Hamiltonian structures can be
naturally expressed making use of the completeness relation and the recursion
operator, whose eigenfunctions are the squared solutions.
