Communications of the Korean Mathematical Society (대한수학회논문집)
- Volume 10 Issue 1
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- Pages.235-249
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- 1995
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- 1225-1763(pISSN)
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- 2234-3024(eISSN)
ON THE M-SOLUTION OF THE FIRST KIND EQUATIONS
- Rim, Dong-Il (Department of Mathematics, College of Natural Sciences, Chungbuk National University) ;
- Yun, Jae-Heon (Department of Mathematics, College of Natural Sciences, Chungbuk National University) ;
- Lee, Seok-Jong (Department of Mathematics, College of Natural Sciences, Chungbuk National University)
- Published : 1995.01.01
Abstract
Let K be a bounded linear operator from Hilbert space $H_1$ into Hilbert space $H_2$. When numerically solving the first kind equation Kf = g, one usually picks n orthonormal functions $\phi_1, \phi_2,...,\phi_n$ in $H_1$ which depend on the numerical method and on the problem, see Varah [12] for more details. Then one findes the unique minimum norm element $f_M \in M$ that satisfies $\Vert K f_M - g \Vert = inf {\Vert K f - g \Vert : f \in M}$, where M is the linear span of $\phi_1, \phi_2,...,\phi_n$. Such a solution $f_M \in M$ is called the M-solution of K f = g. Some methods for finding the M-solution of K f = g were proposed by Banks [2] and Marti [9,10]. See [5,6,8] for convergence results comparing the M-solution of K f = g with $f_0$, the least squares solution of minimum norm (LSSMN) of K f = g.