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

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983562-9

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 ${k_{{x_i}}}$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.

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Keywords:
First-kind integral equation,
numerical solution of first-kind integral equations

Article copyright:
© Copyright 1989
American Mathematical Society