Lower estimates on the condition number of a Toeplitz sinc matrix and related questions

Abstract

As one new result, for a symmetric Toeplitz $ \operatorname{sinc} $ $n \times n$-matrix $A(t)$ depending on a parameter $t$, lower estimates (tending to infinity as t vanishes) on the pertinent condition number are derived. A further important finding is that prior to improving the obtained lower estimates it seems to be more important to determine the lower bound on the parameter $t$ such that the smallest eigenvalue $\mu_n(t)$ of $A(t)$ can be reliably computed since this is a precondition for determining a reliable value for the condition number of the Toeplitz $ \operatorname{sinc} $ matrix. The style of the paper is expository in order to address a large readership.