An analysis of the numerical solution of Fredholm integral equations of the first kind

Summary This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT_Nx=y_N,N=1,2, .... The operatorsT andT_N can be viewed as operators from eitherL₂[a, b] toL₂[c,d] or as operators fromL[a, b] toL[c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT_Nx=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.