Numerical approximation of minimum norm solutions of for special

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be continuous and linear and assume . Define by . Assume *K* has the property that (a) for all and (b) if -a.e., then for all . For example, if , is Lebesgue measure and if satisfies a Lipschitz condition in *x*, then *K* has the above property. Assume *K* satisfies this property and is a minimum norm solution of the first-kind integral equation for all . It is shown that is the -norm limit of linear combinations of the 's. It is then shown how to choose constants to minimize *without knowing what* *is*. This paper also contains results on how to choose the 's as well as numerical examples illustrating the theory.

**[1]**R. S. Anderssen & P. M. Prenter, "A formal comparison of methods proposed for the numerical solution of first kind integral equations,"*J. Austral. Math. Soc. Ser. B*, v. 22, 1981, pp. 488-500. MR**626939 (82h:65094)****[2]**C. W. Groetsch, "Uniform convergence of regularizaron methods for Fredholm equations of the first kind,"*J. Austral. Math. Soc. Ser. A*, v. 39, 1985, pp. 282-286. MR**796038 (86j:45026)****[3]**Glenn R. Luecke & Kevin R. Hickey, "A note on the filtered least squares minimal norm solution of first kind equations,"*J. Math. Anal. Appl.*, v. 100, no. 2, 1984, pp. 635-641. MR**743345 (85k:45022)****[4]**Glenn R. Luecke & Kevin R. Hickey, "Convergence of approximate solutions of operator equations,"*Houston J. Math.*, v. 11, no. 3, 1985, pp. 345-353. MR**808651 (87a:47019)****[5]**Manfred R. Trummer, "A method for solving ill-posed linear operator equations,"*SIAM J. Numer. Anal.*, v. 21, no. 4, 1984, pp. 729-737. MR**749367 (85m:65051)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65R20,
47A50,
65J10

Retrieve articles in all journals with MSC: 65R20, 47A50, 65J10

Additional Information

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

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

Article copyright:
© Copyright 1989
American Mathematical Society