On the approximation numbers of large Toeplitz matrices

Abstract

The $k$th approximation number $s_k^{(p)}(A_n)$ of a complex $n\times n$ matrix $A_n$ is defined as the distance of $A_n$ to the $n\times n$ matrices of rank at most $n-k$. The distance is measured in the matrix norm associated with the $l^p$ norm $(1<p<\infty )$ on ${\mathbf{C}}^n$. In the case $p=2$, the approximation numbers coincide with the singular values.