Numerical approximation of minimum norm solutions of $Kf=g$ for special $K$
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- by Glenn R. Luecke PDF
- Math. Comp. 53 (1989), 563-569 Request permission
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 ${k_{{x_i}}}$βs. It is then shown how to choose constants ${c_1}, \ldots ,{c_n}$ to minimize $\left \| {{f_0} - \sum \nolimits _{j = 1}^n {{c_j}{k_{{x_j}}}} } \right \|$ without knowing what ${f_0}$ is. This paper also contains results on how to choose the ${k_{{x_j}}}$βs as well as numerical examples illustrating the theory.References
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Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 563-569
- MSC: Primary 65R20; Secondary 47A50, 65J10
- DOI: https://doi.org/10.1090/S0025-5718-1989-0983562-9
- MathSciNet review: 983562