On the asymptotic distribution of the eigenvalues of discretizations of a compact operator

Abstract

It is well known that bounds on the asymptotic decay rate of the eigenvalues of the symmetric integral operator with kernel k(s,t) may be obtained from the smoothness of k(s,t) [1,4,7,9,11]. Some recent numerical results suggested that these bounds also applied to the eigenvalues of matrices with entries $K_{mn} = k(s_m,t_n)$ that occur in various discretizations of the continuous operator. This note shows that, in certain common situations, this is indeed the case.