A bound for the remainder of the Hilbert-Schmidt series and other results on representation of solutions to the functional equation of the second kind with a self-adjoint compact operator as an infinite series

Abstract

For the functional equation of the second kind (see (1)) ø − λbdK ø = f, with K a compact self-adjoint linear operator on a Hilbert space (a Fredholm integral equation of the second kind, for example), a bound for the remainder of the Hilbert-Schmidt series is found. It is shown that the series solution to (1) introduced in the author's previous paper [1] is (much) more rapidly convergent than the Hilbert-Schmidt series and generally speaking, is a preferable way of expressing the solution to (1) for regular λ as an infinite series. Other series solutions to (1) are given. The corresponding expressions for the inverse (I − λK) −1 and the resolvent B λ, and also for the resolvent of the Fredholm integral equation of the second kind with symmetric kernel, are given too.