Explicit formulas for the inverses of Toeplitz matrices, with applications

We derive explicit formulas for the inverses of truncated block Toeplitz matrices that have a positive Hermitian matrix symbol with integrable inverse. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the symbol. The derivation of the formulas involves the dual process of a stationary process that has the symbol as spectral density. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of Toeplitz systems. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices that have a rational symbol. The significance of the closed-form formulas is that they provide us with a linear-time algorithm to compute the solutions of corresponding Toeplitz systems.