The numerical solution of Fredholm integral equations of the first kind
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A convergence theorem for Lee and Prenter's filtered leastsquares method for solving the Fredholm first kind equationKf = g is corrected. Under suitable restrictions, the filtered least squares method is shown to be well-posed under compact perturbations in K and arbitrary perturbations in g;The M-solution of the Fredholm first kind equation Kf = g is the unique minimum norm element f (epsilon) M which minimizes (VBAR)(VBAR)Kf - g(VBAR)(VBAR), where M is a finite dimensional subspace of the given Hilbert space. Several convergence results are proved for the M-solution. A modified Gram-Schmidt method for calculating the M-solution of Kf = g is compared to a modification of the normal equations method which is used to calculate the M-solution of Kf = g. This modified Gram-Schmidt method is shown to be well-posed under compact perturbations in K and arbitrary perturbations in g;A convergence theorem for Marti's method for solving the Fredholm first kind equation Kf = g is corrected. Marti's method is shown to be well-posed under perturbations in g. It is shown that Marti's method is not necessarily well-posed under compact perturbations in K. Also included are results relating Marti's solution to the M-solution of Kf = g and to the least square solution of minimum norm of Kf = g;References;Lee, J. W; and Prenter, P. M. "An Analysis of the Numerical Solution of Fredholm Integral Equations of the First Kind." Numerische Mathematik 30 (1978):1-23;Marti, J. T. "An Algorithm for Computing Minimum Norm Solutions of Fredholm Integral Equations of the First Kind." SIAM Journal of Numerical Analysis 15 (December 1978):1071-1076.