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：本文针对多元长记忆平稳过程的逆块Toeplitz矩阵导出了精确的逼近误差界，并分别针对列和与行和两种情形评估了该误差界。这些结果被用于证明一般块Toeplitz系统解的强收敛性。证明的关键环节在于对由Toeplitz矩阵奇异符号所关联的相位函数的傅里叶系数构成的求和项进行界限估计。