Prediction from part of the past of a stationary process (Q794058)

Let w(t) be the spectral density of a stationary process X(t). Denote by Z the space of measurable functions which are square-integrable with the weight w(t), and let Z(a,b) denote the closed subspace of Z generated by \(\{e^{itx}:a\leq t\leq b\}\). The problem of approximating orthogonal projections on Z(-a,a) is considered in the article. Under some assumptions \(Z(-a,a)=Z(-a,\infty)\cap Z(-\infty,a)\), so the projection may be approximated by projecting on Z(-a,\(\infty)\) and Z(-\(\infty,a)\). How good this scheme is, depends upon structural properties of the weight w(t). This problem is discussed in the article.