Abstract
Let k(.,.) be a continuous kernel defined on Omega x Omega, Omega compact subset of R-d, d >= 1, and let us consider the integral operator (K) over tilde from C(Omega) into C(Omega) (C(Omega) set of continuous functions on Omega) defined as the map
f(x) -> l(x) = integral(Omega) k(x,y) f(y) dy, x is an element of Omega
(K) over tilde is a compact operator and therefore its spectrum forms a bounded sequence having zero as unique accumulation point. Here, we first consider in detail the approximation of (K) over tilde by using rectangle formula in the case where Omega = [0,1], and the step is h = 1/n. The related linear application can be represented as a matrix A(n) of size n. In accordance with the compact character of the continuous operator, we prove that {A(n)}similar to(sigma)0 and {A(n)}similar to lambda 0, that is, the considered sequence has singular values and eigenvalues clustered at zero. Moreover, the cluster is strong in perfect analogy with the compactness of (K) over tilde. Several generalizations are sketched, with special attention to the general case of pure sampling sequences, and few examples and numerical experiments are critically discussed, including the use of GMRES and preconditioned GMRES for large linear systems coming from the numerical approximation of integral equations of the form
((I - (K) over tilde) f(t))(x) = g(x), x is an element of Omega, (1)
with ((K) over tilde f(t))(x) = integral(Omega) k(x,y) f(y) dy and datum g(x). Copyright (c) 2014 John Wiley & Sons, Ltd.