We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.

翻译：我们为拖长的托普利茨区块矩阵的反转得出新的明确公式,该公式与多变最低固定过程相对应。公式的主要成分是该过程光谱密度所附带的阶段函数的Fourier系数。公式的衍生依据是适用于固定过程的双重过程的最近开发的有限预测理论。我们用两个应用来说明公式的效用。第一个应用是整个托普利茨区块系统用于多变短模过程的解决方案的强烈趋同结果。第二个应用是相对于多变的ARMA进程的Teplitz区块矩阵反转的封闭式公式。后者的意义在于它们为我们提供了一种线性时间算法,用以计算托普利茨区块系统的相应解决方案。