On the Elementwise Convergence of Continuous Functions of Hermitian Banded Toeplitz Matrices

• Published in: IEEE transactions on information theory Vol. 53; no. 3; pp. 1168 - 1176
• Main Authors: Crespo, P.M., Gutierrez-Gutierrez, J.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2007; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Asymptotic properties; Circulant matrices; Convergence; covariance matrices; Covariance matrix; Educational programs; Eigenvalues and eigenfunctions; elementwise convergence; Exact sciences and technology; functions of matrices; Gain; H infinity control; Information theory; Information, signal and communications theory; Inverse; Mathematical analysis; Mathematical model; Mathematical models; Matrices; Matrix; Matrix methods; Mean square error methods; minimum mean square error (MMSE); Signal processing; stationary stochastic time series; Stochastic processes; Szegö's theorem; Telecommunications and information theory; Toeplitz matrices; minimum mean square error (MMSE); stationary stochastic time series; covariance matrices; Hermite interpolation; Asymptotic behavior; Time series; Theoretical study; functions of matrices; Space time; Continuous function; Invariance; Toeplitz matrix; Modeling; Mean square error; Szegö's theorem; Covariance; elementwise convergence; Circulant matrices; Toeplitz matrices
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2006.890697

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 modeling of problems where some kind of shift invariance occurs in terms of space or time. R. M. Gray's excellent tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szegouml 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. Using similar mathematical tools and to gain insight into the solutions of this sort of problems, the present correspondence derives new theoretical results regarding the convergence of these entries