On moment-discretization and least-squares solutions of linear integral equations of the first kind |
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Authors: | MZ Nashed |
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Institution: | School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 USA |
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Abstract: | Let K(s, t) be a continuous function on 0, 1] × 0, 1], and let be the linear integral operator induced by the kernel K(s, t) on the space 20, 1]. This note is concerned with moment-discretization of the problem of minimizing 6Kx?y6 in the 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝01K(si, t) x(t) dt = y(si), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating (where 2 is the generalized inverse of ), without recourse to the normal equation , that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind. |
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