The numerical treatment of Fredholm integral equations is an important topic. One general approach is to approximate the integral equation with a linear system of equations that generally has a dense and large matrix. This provides a finite-dimensional problem, and makes the construction of a table of the calculated solution possible.

Often, the integral is approximated by using a mechanical quadrature rule, namely, a finite sum. The authors of this work instead study the approximation of the integral operator with an operator of finite rank, whereupon an approximate solution is obtained after solving a linear system of equations. One possibility is to expand the kernel function in a Taylor series, but the authors show that it is better to use interpolation based on Lagrange’s formula. The authors discuss the stability and convergence of this approach, and also address the cases of multidimensional kernels, as well as some important classes of singular kernels. General results are established. A model problem is analyzed in detail, but no numerical results are presented.

The presentation is clear, and should be accessible to a large audience. The reference list is adequate, and contains seven items.