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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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 ${k_{{x_i}}}$’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 ${k_{{x_j}}}$’s as well as numerical examples illustrating the theory.
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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