On the relation between Green’s functions and covariances of certain stochastic processes and its application to unbiased linear prediction

Abstract

As a result of the stimulus of the work of Wiener [1](2) and Kolmogoroff [2] the theory of linear prediction has developed during the past decade. In this paper we shall develop a method of solution for a class of problems which are natural generalizations of these that have been treated by the socalled auto-correlation theory as developed by Phillips and Weiss [3] or Cunningham and Hynd [4]. A basic difficulty in a linear prediction theory employing continuous observations appears to be that of solving the integral equations of the first kind that arise from the minimization of the error variance. Since, as in the auto-correlation theory, the prediction is based on a finite past history, such tools as Wiener's generalized Fourier analysis are not available and a general method of solution of the integral equations has not yet been devised. Here we develop a method of solution that requires hypotheses which are satisfied in many important physical applications. Moreover, as will be seen from Part I of the present paper, the kernel of our method is based on the intrinsically interesting relationship that exists between covariance functions of random processes generated by driving nth order linear differential equations by so-called pure noise and the Green's function of a suitably defined self-adjoint equation of order 2n. For physically stable linear differential equations with constant coefficients, it will be shown (Corollary 1.1) that such covariance functions are in fact Green's functions of a suitably defined self-adjoint problem. For linear differential equations with variable coefficients this is no longer true but such covariance