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International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 23-26
http://dx.doi.org/10.1155/S0161171288000055

Remarks on extreme eigenvalues of Toeplitz matrices

Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, USA

Received 11 April 1986

Copyright © 1988 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f be a nonnegative integrable function on [π,π], Tn(f) the (n+1)×(n+1) Toeplitz matrix associated with f and λ1,n its smallest eigenvalue. It is shown that the convergence of λ1,n to minf(0) can be exponentially fast even when f does not satisfy the smoothness condition of Kac, Murdoch and Szegö (1953). Also a lower bound for λ1,n corresponding to a large class of functions which do not satisfy this smoothness condition is provided.