Projection methods for equations of the second kind

A class of projection methods, differing from the classical projection methods, is studied for the equation y = f + Ky, where K is a compact linear operator in a Banach space E, and f?E, In these methods K is approximated by a finite-rank operator K_n, which is constructed with the aid of certain projection operators, and which satisfies K_nz = Kz for all z belonging to a chosen subspace U_n ? E. Under certain conditions, it is shown that the convergence of the approximate solution is faster than that of any classical projection method based on the subspace U_n. In an example, U_n is taken to consist of piecewise constant functions, and the projections are so chosen that the method becomes equivalent to a single iteration of a classical method, the collocation method; in this case the error (in the supremum norm) is $\text{O(}\text{1}\text{n}{}^2\text{)}$, compared with $\text{O(}\text{1}\text{n}\text{)}$ for the collocation method.