Block-Toeplitz matrices and associated properties of a Gaussian model on a half-axis

A Gaussian model on a half-axis with interaction given by a block-Toeplitz matrix $\{s_{j-k}\}^\infty_{j,k=0}$. is studied. A procedure is indicated for calculating the correlation functions and the free energy in the absence of an external field and for several ways of including such a field. The results are formulated in terms of a matrix measure $\sigma$, whose Fourier coefficients are $s_j$. These results are based on the asymptotic behavior found in the paper for the individual blocks of the matrix $(\{s_{j-k}\}^n_{j,k=0})^{-1}$ and their sums in the limit $n\to\infty$.