An explicit formula for the inverse of band triangular Toeplitz matrix

In order to estimate the condition number of the preconditioned matrix proposed in [F.R. Lin, W.K. Ching, Inverse Toeplitz preconditioners for Hermitian Toeplitz systems, Numer. Linear Algebra Appl. 12 (2005) 221–229], we study the inverse of band triangular Toeplitz matrix. We derive an explicit formula for the entries of the inverse of band lower triangular Toeplitz matrix by means of divided difference and use the formula to estimate the condition number of the preconditioned matrices. In particular, we prove that the minimal eigenvalue of preconditioned matrix is well separated from the origin.