Singular values of Cauchy-Toeplitz matrices

Abstract The behavior of singular values of matrices A n =[1/( i − j + g )] n i , 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 e-neighborhood of the clustering point [ o ( n )/ n →0 as n →∞]; o ( n ) = O(ln 2 n ) holds if ⋎ g ⋎ ⩽ 1 2 . Also proved is that the minimum singular values of A n ( g ) tend to zero provided that ⋎ g ⋎⩾ 1 2 .