Singular values of cauchy-toeplitz matrices

AbstractThe behavior of singular values of matrices An =[1/(i−j+g)]ni,j=1 with n→∞ is investigated. For any real g which is not integer it is proved that the singular values are clustered at π / ⋎sin π g⋎, which is their upper boundary. The only o(n) singular values are those which lie outside a given ε-neighborhood of the clustering point [o(n)/n→0 as n→∞]; o(n) = O(ln2n) holds if ⋎g⋎ ⩽12. Also proved is that the minimum singular values of An(g) tend to zero provided that ⋎g⋎⩾12