{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:15Z","timestamp":1787329275993,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>After a brief survey of basic results about finite-dimensional approximation settings (such as mathematical frameworks of various projection methods, including the least-squares method, the dual least-squares method, and the Galerkin method) for infinite-dimensional Moore\u2013Penrose inverses, this paper proceeds to a detailed study from the following aspects: For projection methods, we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite-dimensional Moore\u2013Penrose inverses; this investigation yields a fundamental convergence theorem (Theorem 2.2), from which the criterion for convergence, the criterion for weak convergence, and the generalized dual least-squares method are derived. We also derive general results on the least-squares method, by which two flaws in Groestch's results are corrected. For nonprojection methods (whose approximation setting is a more general framework), we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding (Theorem 3.2). Several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior.<\/jats:p>","DOI":"10.1137\/060661120","type":"journal-article","created":{"date-parts":[[2008,4,4]],"date-time":"2008-04-04T18:08:07Z","timestamp":1207332487000},"page":"1454-1482","source":"Crossref","is-referenced-by-count":12,"title":["Finite-Dimensional Approximation Settings for Infinite-dimensional Moore\u2013Penrose Inverses"],"prefix":"10.1137","volume":"46","author":[{"given":"Nailin","family":"Du","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,4]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"K. Atkinson and W. Han,\n                      Theoretical Numerical Analysis\n                      , Springer-Verlag, New York, 2001.","DOI":"10.1007\/978-0-387-21526-6"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)00210-X"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00509-5"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/01630560500377329"},{"key":"R5","first-page":"43","volume":"50","author":"Du N. L.","year":"2007","journal-title":"Acta Math. Sinica"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"H. W. Engl,\n                      On the convergence of regularization methods for ill-posed linear operator equations\n                      , in Improperly Posed Problems and Their Numerical Treatment, G. H\u00e4mmerlin and K. H. Hoffmann, eds., Birkh\u00e4user, Basel, 1983, pp. 81\u201395.","DOI":"10.1007\/978-3-0348-5460-3_6"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"H. W. Engl, M. Hanke, and A. Neubauer,\n                      Regularization of Inverse Problems\n                      , Kluwer, Dordrecht, 2000.","DOI":"10.1007\/978-94-009-1740-8_3"},{"key":"R8","first-page":"17","volume":"17","author":"Engl H. W.","year":"1988","journal-title":"Proc. Centre Math. Anal. (Canberra)"},{"key":"R9","first-page":"173","volume":"4","author":"Groetsch C. W.","year":"1982","journal-title":"J. Integral Equations"},{"key":"R10","unstructured":"C. W. Groetsch,\n                      The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind\n                      , Pitman, London, 1984."},{"key":"R11","unstructured":"C. W. Groetsch,\n                      Generalized Inverses of Linear Operators: Representation and Approximation\n                      , Dekker, New York, 1977."},{"key":"R12","first-page":"201","volume":"14","author":"Groetsch C. W.","year":"1988","journal-title":"Houston J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0362-1588","issn-type":"print"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(89)90019-1"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(83)90133-8"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02247939"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"A. Kirsch,\n                      An Introduction to the Mathematical Theory of Inverse Problems\n                      , Springer-Verlag, New York, 1996.","DOI":"10.1007\/978-1-4612-5338-9"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko,\n                      Approximate Solution of Operator Equations\n                      , translated by D. Louvish, Wolters-Noordhoff Publishing, Groningen, 1972.","DOI":"10.1007\/978-94-010-2715-1"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"R. Kress,\n                      Linear Integral Equations\n                      , Springer-Verlag, Berlin, 1989.","DOI":"10.1007\/978-3-642-97146-4"},{"key":"R19","first-page":"17","volume":"99","author":"Koliha J. J.","year":"1999","journal-title":"Math. Proc. R. Ir. Acad."},{"key":"R20","first-page":"1294","volume":"33","author":"Ma J. P.","year":"1990","journal-title":"$A^{\\dag }$, Sci. China Ser. A"},{"key":"R21","first-page":"251","volume":"13","author":"Ma J.","year":"1992","journal-title":"$A^{\\dag }$, Chinese Ann. Math. Ser. B"},{"key":"R22","unstructured":"M. Z. Nashed, ed.\n                      Generalized Inverses and Applications\n                      , Academic Press, New York, 1976."},{"key":"R23","doi-asserted-by":"crossref","unstructured":"M. Z. Nashed,\n                      Perturbations and approximations for generalized inverses and linear operator equations\n                      , in Generalized Inverses and Applications, M. Z. Nashed, ed., Academic Press, New York, 1976, pp. 325\u2013396.","DOI":"10.1016\/B978-0-12-514250-2.50013-5"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386397"},{"key":"R25","first-page":"163","volume":"49","author":"Rako\u010devi\u0107 V.","year":"1997","journal-title":"Mat. Vesnik"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1080\/01630569208816485"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01686719"},{"key":"R28","unstructured":"G. Wang, Y. Wei, and S. Qiao,\n                      Generalized Inverses: Theory and Computations\n                      , Science Press, Beijing, 2004."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(99)00271-4"},{"key":"R30","unstructured":"K. Yosida,\n                      Functional Analysis\n                      , 6th ed., Springer-Verlag, Berlin, 1980."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060661120","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:43Z","timestamp":1787325223000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060661120"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060661120"],"URL":"https:\/\/doi.org\/10.1137\/060661120","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}