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.

**[1]**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**0051437**, https://doi.org/10.1090/S0002-9947-1950-0051437-7**[2]**J. B. Diaz and F. T. Metcalf,*On iteration procedures for equations of the first kind, 𝐴𝑥=𝑦, and Picard’s criterion for the existence of a solution*, Math. Comp.**24**(1970), 923–935. MR**0281376**, https://doi.org/10.1090/S0025-5718-1970-0281376-4**[3]**Richard J. Hanson,*A numerical method for solving Fredholm integral equations of the first kind using singular values*, SIAM J. Numer. Anal.**8**(1971), 616–622. MR**0293867**, https://doi.org/10.1137/0708058**[4]**E. Hellinger & O. Toeplitz, "Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten,"*Enzyklopädie der Mathematischen Wissenschaften*II C13, 1928, p. 1349.**[5]**W. J. Kammerer and M. Z. Nashed,*Iterative methods for best approximate solutions of linear integral equations of the first and second kinds*, J. Math. Anal. Appl.**40**(1972), 547–573. MR**0320677**, https://doi.org/10.1016/0022-247X(72)90002-9**[6]**M. Z. Nashed,*Generalized inverses, normal solvability, and iteration for singular operator equations*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 311–359. MR**0275246****[7]**M. Z. Nashed & G. Wahba,*Approximate Regularized Solutions to Linear Operator Equations When the Data-Vector is Not in the Range of the Operator*, MRC Technical Summary Report #1265, Mathematics Research Center, The University of Wisconsin, 1973.**[8]**B. Noble,*A Bibliography on: "Methods for Solving Integral Equations"--Subject Listing*, MRC Technical Summary Report #1177, Mathematics Research Center, The University of Wisconsin, 1971.**[9]**W. V. Petryshyn,*On generalized inverses and on the uniform convergence of (𝐼-𝛽𝐾)ⁿ with application to iterative methods*, J. Math. Anal. Appl.**18**(1967), 417–439. MR**0208381**, https://doi.org/10.1016/0022-247X(67)90036-4**[10]**H. L. Shapiro,*Topics in Approximation Theory*, Lecture Notes in Math., vol. 187, Springer-Verlag, Berlin, 1970.**[11]**David W. Showalter and Adi Ben-Israel,*Representation and computation of the generalized inverse of a bounded linear operator between Hilbert spaces*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**48**(1970), 184–194 (English, with Italian summary). MR**0273432****[12]**F. Smithies,*Integral equations*, Cambridge Tracts in Mathematics and Mathematical Physics, no. 49, Cambridge University Press, New York, 1958. MR**0104991****[13]**O. N. Strand,*Theory and Methods for Operator Equations of the First Kind*, Ph.D. Thesis, Colorado State University, Fort Collins, Colorado, 1972, 86 pp.**[14]**Grace Wahba,*Convergence rates of certain approximate solutions to Fredholm integral equations of the first kind*, J. Approximation Theory**7**(1973), 167–185. MR**0346453****[15]**Grace Wahba,*A class of approximate solutions to linear operator equations*, J. Approximation Theory**9**(1973), 61–77. MR**0386284****[16]**G. Wahba,*Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations*, MRC Technical Summary Report #1132, Mathematics Research Center, The University of Wisconsin, 1973.

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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