Diagonalization of the auto-correlation functions of random signals in terms of Cauchy wavelets using operator algebra

The eigenfunction system of a kind of operator is identical to the over-complete Cauchy wavelet system, as has been shown previously. In this presentation, the re-ordering problem among this operator and its adjoint is investigated. Then, an application of this re-ordering problem to the pseudo-diagonalization problem of the autocorrelation function of a stochastic process is proposed.