The projection method is a well-known approach to numerically solving singular Fredholm integral equations of the second kind. The projection spaces can be either piecewise polynomials or globally smooth functions, typically eigenfunctions of Sturm-Liouville problems.

The method analyzed in this paper is a collocation projection, where the approximate solution is a linear combination of orthogonal Jacobi polynomials and the collocation points are the roots of the m-th Jacobi polynomial. The main result is that the error is of almost optimal order log m/m^r+1 and that the condition of the linear system is order log m. Here, r is the regularity of the right-hand side. The point of the analysis is that the error is measured in the uniform norm, which has been an open problem so far.

For those cases in which the kernel of the integral equation is smooth, a fully discrete collocation method is introduced that has similar convergence properties. The method is also applied to Cauchy singular integral equations, using a regularization procedure that the authors introduced in previous work.