Asymptotic behavior of the condition number of two-level Toeplitz matrix sequences (Q1765908)

The main objective of this paper is the analysis of the Euclidean conditioning \(k\left( \cdot \right) \) of two-level Toeplitz matrices of the form \(T_{nm}\left( f\right) \) where \(n\) and \(m\) are large and the symbol \(f\) is assumed to be \(L^{\infty }\) over \(I^{2}=\left( -\pi ,\pi \right] ^{2}\) and weakly sectorial. It is shown that the upper bound of the condition number will depend on the maximal order of the essential zeros of \(\text{Re}f \); the presence of large \(\text{Im}f\) will decrease the condition number.