An algorithm for spectral factorization using random search techniques

An approximation method for the energy spectrum of a stationary stochastic dynamical system is presented, which allows approximate functional rational factorization.This paper is in three parts. The first deals with a theoretical problem of approximation in Hardy Spaces, whose main result is the following:Let S⁽ⁱⁿ⁾, S be positive functions belonging to L¹(R_gw), such that log S⁽ⁿ⁾ and log S belong to L¹(R_ω, dω/1 + ω²).Let h⁽ⁿ⁾, h be the outer functions of the Hardy Space H²(Π₊) such that S^(g) = |h⁽ⁿ⁾|² and S = |h|² on iR.If S⁽ⁿ⁾ n^→∝ S in L¹(R_ω), and log S⁽ⁿ⁾ n^→∝ log S in L¹(R_ω, dω/1 + ω²), then: h⁽ⁿ⁾ n^→∝ h in H²(Π₊).The second part describes an effective algorithm, using random search methods, and gives an almost sure convergence result for it.The third part treats numerically two examples, permitting comparison of this algorithm with others (whenever there are…): the first example is a problem of approximation for a nonrational process (turbulence) that was considered in Ref. 22: the second example is a problem of model reduction (automatic) considered in Ref. 4.