Block-Toeplitz matrices and associated properties of a Gaussian model on a half-axis
Let \(\{s_ j\}^{+\infty}_{-\infty}\) be a sequence of \(m\times m\) matrices (it describes in the paper the interaction in a Gaussian model on a half-axis). Suppose that \(S_ n=(s_{j-k})^ n_{j,k=0}>0\), \(n=1,2,... \). Split the matrix \(S_ n^{-1}\) into \(m\times m\) blocks: \(S_ n^{-1}=(v_{jk}^{(n)})\). The paper contains proofs of the expressions for \(\lim_{n}v_{jk}^{(n)}=v_{jk}\) and \(\lim_{n}v_ n=v\), where \(v_ n=\sum^{n}_{i,j=0}v_{jk}^{(n)}.\) Let us describe the first result (Theorem 1) of the paper. If \(s_ j=(2\pi)^{-1}\int^{\pi}_{-\pi}e^{-ij\theta}d\sigma (\theta)\), where the measure \(\sigma\) has a decomposition into parts \(d\sigma (\theta)=G(\theta)d\theta +d\sigma_ s(\theta)\) where ln det \(G\in L^ 1\) and \(\sigma_ s\) is singular with respect to the Lebesgue measure then \(G(\theta)=F_+(e^{i\theta})F_+(e^{i\theta})^*\) a.e. and denoting by \(g_ j\) the Taylor coefficients of \(F_+^{-1}\) one has: Theorem. Under the above conditions the lim \(v_{jk}\) exists and \(v_{jk}=\sum^{\min (j,k)}_{r=0}g^*_{j-r}g_{k-r}\).
scientific article; zbMATH DE number 3983924