# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Numerical approximation of minimum norm solutions of $Kf=g$ for special $K$HTML articles powered by AMS MathViewer

by Glenn R. Luecke
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 ${k_{{x_i}}}$βs. 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 ${k_{{x_j}}}$βs as well as numerical examples illustrating the theory.
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