ABSTRACT
This chapter provides some insight into the connection between numerical methods, the aspects of practical approximation and functional-analytic considerations. It examines two typical methods of determining approximate solutions of Fredholm integral equations of the second kind: the method of applying a quadrature formula to the integral equation and the method of substitution kernels, particularly degenerate substitution kernels which are constructed using several types of splines. The chapter also examines the homogeneous Fredholm integral equation of the second kind, which represents the eigenvalue problem. Several numerical examples have been computed in order to compare the efficiency of different substitution kernel methods, and, especially, the convergence order predicted by the bounds.
