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Mathematics of Computation

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Numerical approximation of minimum norm solutions of $ Kf=g$ for special $ K$

Author: Glenn R. Luecke
Journal: Math. Comp. 53 (1989), 563-569
MSC: Primary 65R20; Secondary 47A50, 65J10
MathSciNet review: 983562
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Abstract: Let $ K:{L_2}(Y,\mu ) \to {L_2}(X,\nu )$ be continuous and linear and assume $ (Kf)(x) = \smallint_Y^{} {k(x,y)f(y)\,d\mu (y)} $. Define $ {k_x}$ by $ {k_x}(y) = k(x,y)$. Assume K has the property that (a) $ {k_x} \in {L_2}(Y,\mu )$ for all $ x \in X$ and (b) if $ Kf = 0\;\nu $-a.e., then $ (Kf)(x) = 0$ for all $ x \in X$. For example, if $ X = Y = [0,1]$, $ \mu = \nu $ is Lebesgue measure and if $ k(x,y)$ satisfies a Lipschitz condition in x, then K has the above property. Assume K satisfies this property and $ {f_0}$ is a minimum $ {L_2}$ norm solution of the first-kind integral equation $ (Kf)(x) = g(x)$ for all $ x \in X$. It is shown that $ {f_0}$ is the $ {L_2}$-norm limit of linear combinations of the $ {k_{{x_i}}}$'s. It is then shown how to choose constants $ {c_1}, \ldots ,{c_n}$ to minimize $ \left\Vert {{f_0} - \sum\nolimits_{j = 1}^n {{c_j}{k_{{x_j}}}} } \right\Vert$ without knowing what $ {f_0}$ is. This paper also contains results on how to choose the $ {k_{{x_j}}}$'s as well as numerical examples illustrating the theory.

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Keywords: First-kind integral equation, numerical solution of first-kind integral equations
Article copyright: © Copyright 1989 American Mathematical Society

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