Eigenvalue Problems for Exponential-Type Kernels

We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate |lambda(k) - lambda(k,h)| = C(k)h(2) in the case of lowest order approximation for both Galerkin and Nystrom methods, where h is the mesh size, lambda(k) and lambda(k, h) are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that vertical bar lambda((G))(k, h) - lambda(k, h) ((N)) vertical bar <= = Ch(2), where lambda((G))(k,h) and lambda((N))(k,h) denote the kth largest eigenvalues computed by Galerkin and Nystrom methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.