We derive sharp approximation error bounds for inverse block Toeplitz matrices associated with multivariate long-memory stationary processes. The error bounds are evaluated for both column and row sums. These results are used to prove the strong convergence of the solutions of general block Toeplitz systems. A crucial part of the proof is to bound sums consisting of the Fourier coefficients of the phase function attached to the singular symbol of the Toeplitz matrices.

翻译：我们为多变量长期记忆平稳过程相关的不完全Toeplitz方块矩阵的逆推导出了尖锐的逼近误差界限。误差边界用于两种列和行之和都可量化的情况。这些结果被用于证明一般块Toeplitz系统的解的强收敛性。证明的一个关键部分是限制由连接到Toeplitz矩阵的奇异符号的相位函数的傅里叶系数构成的总和。