A Note on the Elementwise Convergence of Functions of Toeplitz Matrices

Toeplitz matrices and functions of Toeplitz matrices (such as the inverse of a Toeplitz matrix, powers of a Toeplitz matrix or the exponential of a Toeplitz matrix) arise in many different theoretical and applied fields. They can be found in the mathematical modelling of problems where some kind of shift invariance occurs in terms of space or time. For the engineering community, R. M. Gray\´s tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego distribution theory on the asymptotic behavior of continuous functions of Toeplitz matrices. His asymptotic results, widely used in engineering due to the simplicity of its mathematical proofs, do not concern individual entries of these matrices but rather, they describe an "average" behavior. However, there are important applications where the asymptotic expressions of interest are directly related to the convergence of a single entry of a continuous function of a Toeplitz matrix. To gain insight into the solutions of this sort of problems, the present paper introduces new theoretical results regarding the convergence of these entries, that can be derived by using similar mathematical tools.