Effective Solutions of Linear Approximation Problems for Multivariate Stationary Processes with a Rational Spectrum

We consider a class of multivariate stationary random processes $\xi(t)=\{\xi_1(t),\dots,\xi_k(t)\}$ having the nonsingular spectral density matrix $||f_{jk}(\lambda)||$, where all $f_{jk}(\lambda)$ are rational functions of $\lambda$. The following linear approximation problems for the processes are studied: 1) the simplest extrapolation problem of determining a linear least-square estimate of $\xi_k(t+\tau),\tau>0$, by known values of $\xi_j(t'),j=1, \dots,n,t'\leq t$; 2) the finite interval extrapolation problem of a linear least-square estimation of $\xi_k(t+\tau)$ by $\xi_j(t'),j=1,\dots,n,t-T\leq t'\leq t$; 3) the interpolation problem of a least-square estimation of $\xi_k(t+\tau),0< \tau<T$ by $\xi_j(t'),j=1,\dots,n,t'\leq t$ or $t'\geq t+T$; 4) the filtration problem of a least-square estimation of the value of some random variable $\Xi$ (such that the functions $f_{\Xi k}(\lambda),k=1,\dots,n$, from equations (3.1)–(3.2) have the form (3.4), where all $q_{rk}(\lambda)$ are rational) by the values of $\xi _j(t'),j=1,\dots,n,t'\leq t$ or $t-T\leq t'\leq t$.
In all cases the method used in previous papers [11] and [12] enables the explicit extrapolation, interpolation or filtration formulae to be derived by merely solving the algebraical equation $D(\lambda)=\det||f_{jk}(\lambda)||=0$ and afterwards a simple system of linear algebraical equations. The same method can also be applied to the case when we wish to find a least-square estimate of $\xi_k(t+\tau)$ or $\Xi$ by the values of $\xi_j(t'),j=1,\dots,n$, on any set of closed intervals on the time axis. Some other generalizations of extrapolation, interpolation and filtration problems may be solved by the same method; they are given in the last section of the paper.