Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

M. Z. Nashed, Grace Wahba
1974 Mathematics of Computation  
We consider approximations \xn] obtained by moment discretization to (i) the minimal £2-norm solution of Six = y where 3C is a Hilbert-Schmidt integral operator on £2, and to (ii) the least squares solution of minimal £2-norm of the same equation when y is not in the range 5i(X) of X. In case (i), if y £ X^y, where 3Cf is the generalized inverse of X. and ||a:"|| -> <» otherwise. Rates of convergence are given in this case if further X^y £ 3C*(£2), where X* is the adjoint of 3C, and the
more » ... Schmidt kernel of X3C* satisfies certain smoothness conditions. In case (ii), if y £ (R(3C) © °> otherwise. If further Xry £ 3C*3C(£2), then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of (XX*)2. Some of these results are generalized to a class of linear operator equations on abstract Hubert spaces.
doi:10.1090/s0025-5718-1974-0461895-1 fatcat:wccfvi3hh5eb7of2d6wcmqiz5y