On the diagonal approximation of the auto-correlation function with the wavelet basis which is optimal with respect to the relative entropy

If the covariance function of a random signal can be written in a diagonal form via the wavelet basis, this random signal can be regarded as a superposition of the wavelets which arise randomly. However, it is known that, in general, such an expression is not possible. In this paper, in place of a perfect diagonalization, an optimal approximate diagonalization in the sense of the relative entropy is investigated theoretically. Especially, it is shown that when a set of wavelets forming complete orthonormal sets expressed in a vector form as {φ _i} is used as the basis, an optimal diagonal approximation of the covariance matrix Γ is not the diagonal form Σ_h (φ¯_h^τΓφ_h )φ_hφ¯_h^τ using the so-called `wavelet spectrum´ but Σ_h(φ¯_h^τΓ ^-1φ_h)^-1φ_hφ¯ _h^τ. Further, several examples are given where Haar wavelets are used