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

Authors: M. Z. Nashed and Grace Wahba
Journal: Math. Comp. 28 (1974), 69-80
MSC: Primary 65J05; Secondary 47A50
MathSciNet review: 0461895
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Abstract: We consider approximations $ \{ {x_n}\} $ obtained by moment discretization to (i) the minimal $ {\mathcal{L}_2}$-norm solution of $ \mathcal{K}x = y$ where $ \mathcal{K}$ is a Hilbert-Schmidt integral operator on $ {\mathcal{L}_2}$, and to (ii) the least squares solution of minimal $ {\mathcal{L}_2}$-norm of the same equation when y is not in the range $ \mathcal{R}(\mathcal{K})$ of $ \mathcal{K}$. In case (i), if $ y \in \mathcal{R}(\mathcal{K})$, then $ {x_n} \to {\mathcal{K}^\dag }y$, where $ {\mathcal{K}^\dag }$ is the generalized inverse of $ \mathcal{K}$, and $ \left\Vert {{x_n}} \right\Vert \to \infty $ otherwise. Rates of convergence are given in this case if further $ {\mathcal{K}^\dag }y \in {\mathcal{K}^\ast}({\mathcal{L}_2})$, where $ {\mathcal{K}^\ast}$ is the adjoint of $ \mathcal{K}$, and the Hilbert-Schmidt kernel of $ \mathcal{K}{\mathcal{K}^\ast}$ satisfies certain smoothness conditions. In case (ii), if $ y \in \mathcal{R}(\mathcal{K}) \oplus \mathcal{R}{(\mathcal{K})^ \bot }$, then $ {x_n} \to {\mathcal{K}^\dag }y$, and $ \left\Vert {{x_n}} \right\Vert \to \infty $ otherwise. If further $ {\mathcal{K}^\dag }y \in {\mathcal{K}^\ast}\mathcal{K}({\mathcal{L}_2})$, then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of $ {(\mathcal{K}{\mathcal{K}^\ast})^2}$.

Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.

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Keywords: First kind integral equations, linear operator equations, moment discretization, convergence rates, least squares solutions, generalized inverses, reproducing kernel Hilbert spaces, Picard criterion
Article copyright: © Copyright 1974 American Mathematical Society