Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind

Authors:
M. Z. Nashed and Grace Wahba

Journal:
Math. Comp. **28** (1974), 69-80

MSC:
Primary 65J05; Secondary 47A50

DOI:
https://doi.org/10.1090/S0025-5718-1974-0461895-1

MathSciNet review:
0461895

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Abstract: We consider approximations obtained by moment discretization to (i) the minimal -norm solution of where is a Hilbert-Schmidt integral operator on , and to (ii) the least squares solution of minimal -norm of the same equation when *y* is not in the range of . In case (i), if , then , where is the generalized inverse of , and otherwise. Rates of convergence are given in this case if further , where is the adjoint of , and the Hilbert-Schmidt kernel of satisfies certain smoothness conditions. In case (ii), if , then , and otherwise. If further , then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of .

Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0461895-1

Keywords:
First kind integral equations,
linear operator equations,
moment discretization,
convergence rates,
least squares solutions,
generalized inverses,
reproducing kernel Hilbert spaces,
Picard criterion

Article copyright:
© Copyright 1974
American Mathematical Society