Optimal convergence rates for some discrete projection methods

摘要

Using recent results of Sloan on hyperinterpolation operators [J. Approx. Theory 81 (1995) 238] we show how to obtain optimal convergence rates for a variety of discrete polynomially based Galerkin and collocation methods for a variety of Fredholm and singular integral equations. These results improve on those given in Golberg [J. Integral Equations Appl. 8 (1996) 307] and are analogous to those obtained by Atkinson and Bogomolny for spline-based methods [Math. Comp. 48 (1987) 596]. These results are then applied to analyze double approximation methods where both numerical integration and data errors are considered. Here we show that the errors are dominated by those in the data. Hence, it may not be possible, in practice, to obtain optimal convergence rates unless the data can be computed exactly.