A Course on Optimization and Best Approximation pp 145-233 | Cite as
Selected special topics
Chapter
First Online:
Keywords
Hilbert Space Banach Space Generalize Inverse Extremal Solution Finite Dimensional Subspace
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References for §31
- 1).E. Asplund and R. T. Rockafellar, Gradients of convex functions. Trans. Amer. Math. Soc. 139(1969), 443–467.MathSciNetCrossRefMATHGoogle Scholar
- 2).E. W. Cheney, Introduction to Approximation Theory. McGraw Hill, New York, 1966.MATHGoogle Scholar
- 3).M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Amer. Math. Soc. 47(1941), 313–317.MathSciNetCrossRefMATHGoogle Scholar
- 4).N. Dunford and J. Schwartz, Linear Operators, Part I. Interscience, New York, 1958.MATHGoogle Scholar
- 5).K. Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space. Duke Math. J. 25(1958), 553–568.MathSciNetCrossRefMATHGoogle Scholar
- 6).R. Holmes, Approximating best approximations. Nieuw Arch. voor Wisk. 14(1966), 106–113.MathSciNetMATHGoogle Scholar
- 7)._____ and B. Kripke, Smoothness of approximation, Mich. Math. J. 15(1968), 225–248.MathSciNetCrossRefMATHGoogle Scholar
- 8).C. McCarthy, cp. Israel Math. J. 5(1967), 249–271.CrossRefGoogle Scholar
- 9).E. McShane, Linear functionals on certain Banach spaces. Proc. Amer. Math. Soc. 1(1950), 402–408.MathSciNetCrossRefMATHGoogle Scholar
- 10).C. Morawetz, Two Lp inequalities. Bull. Amer. Math. Soc. 75(1969), 1299–1302.MathSciNetCrossRefMATHGoogle Scholar
- 11).V. Sholohovich, Unstable extremal problems and geometric properties of Banach spaces. Soviet Math. Dokl. 11(1970), 1470–1472.Google Scholar
References for §32
- 1).T. Ando, Contractive projections in Lp spaces. Pac. J. Math. 17(1966), 391–405.CrossRefMATHGoogle Scholar
- 2).I. Daugavet, A property of completely continuous operators in the space C. Uspehi Mat. Nauk 18(1963), 157–158.(Russian)MathSciNetMATHGoogle Scholar
- 3).M. Day, Normed Linear Spaces. Academic Press, New York, 1962.CrossRefMATHGoogle Scholar
- 4).J. Dugundji, Topology. Allyn and Bacon, Boston, 1966.MATHGoogle Scholar
- 5).C. Foias and I. Singer, Points of diffusion of linear operators, Math. Zeit. 87(1965), 434–450.MathSciNetCrossRefMATHGoogle Scholar
- 6).A. Garkavi, Approximative properties of subspaces with finite defect in the space of continuous functions. Sov. Math. 5(1964), 440–443.MathSciNetMATHGoogle Scholar
- 7).R. Holmes, Approximating best approximations. Nieuw Arch. voor Wisk. 14(1966), 106–113.MathSciNetMATHGoogle Scholar
- 8)._____, On the continuity of best approximation operators. Proc. Symp. Inf. Dim. Topology. Annals of Math Study #69, Princeton Univ. Press, to appear.Google Scholar
- 9)._____ and B. Kripke, Smoothness of approximation. Mich. Math. J. 15(1968), 225–248.MathSciNetCrossRefMATHGoogle Scholar
- 10)._____, Best approximation by compact operators. Ind. Univ. Math. J., to appear.Google Scholar
- 11).V. Klee, Two renorming constructions related to a question of Anselone. Studia Math. 23(1969), 231–242.MathSciNetMATHGoogle Scholar
- 12).B. Kripke and T. Rivlin, Approximation in the metric of L1(X,μ). Trans. Amer. Math. Soc. 119(1965), 101–122.MathSciNetMATHGoogle Scholar
- 13).J. Lambert, The weak sequential continuity of the metric projection in Lp spaces. Dissertation, Purdue Univ., 1970.Google Scholar
- 14).A. Lazar, P. Morris, and D. Wulbert, Continuous selections for metric projections. J. Func. Anal. 3(1969), 139–216.MathSciNetCrossRefMATHGoogle Scholar
- 15).J. Lindenstrauss, On nonlinear projections in Banach spaces. Mich. Math. J. 11(1964), 263–287.MathSciNetCrossRefMATHGoogle Scholar
- 16)._____, Extension of compact operators. Mem. Amer. Math. Soc. #48, 1964.Google Scholar
- 17).E. Michael, Selected selection theorems. Amer. Math. Monthly 63(1956), 233–238.MathSciNetCrossRefMATHGoogle Scholar
- 18).R. Moroney, The Haar problem in L1. Proc. Amer. Math. Soc. 12(1961), 793–795.MathSciNetMATHGoogle Scholar
- 19).F. Murray, On complementary manifolds and projections in Lp and ϕp. Trans. Amer. Math. Soc. 41(1937), 138–152.MathSciNetGoogle Scholar
- 20).T. Newman and P. Odell, On the concept of a p-q generalized inverse of a matrix. SIAM J. Appl. Math. 17(1969), 520–525.MathSciNetCrossRefMATHGoogle Scholar
- 21).E. Oshman, On continuity of metric projections onto some classes of subspaces in a Banach space, Sov. Math. 11(1970), 1521–1523.MATHGoogle Scholar
- 22).I. Singer, Best Approximation in Normed Linear Subspaces by Elements of Linear Subspaces. Springer, Berlin-Heidelberg, 1970.CrossRefMATHGoogle Scholar
References for §33
- 1).P. Belobrov, On the problem of the Chebyshev center of a set. Izv. Vys. Ucheb. Zaved. (1964), 3–9. (Russian)Google Scholar
- 2).L. Blumenthal and G. Wahlin, On the spherical surface of smallest radius enclosing a bounded subset of n-dimensional euclidean space. Bull. Amer. Math. Soc. 47 (1941), 771–777.MathSciNetCrossRefMATHGoogle Scholar
- 3).L. Danzer, B. Grünbaum, and V. Klee, Helly’s theorem and its relatives. Convexity, Proc. Symp. Pure Math. 7 (1963), Amer. Math. Soc.; 101–180.CrossRefMATHGoogle Scholar
- 4).M. Day, Normed Linear Spaces. Academic Press, New York, 1962.CrossRefMATHGoogle Scholar
- 5).D. Dean, Direct factors of (AL)-spaces. Bull. Amer. Math. Soc 71 (1965), 368–371.MathSciNetCrossRefMATHGoogle Scholar
- 6).A. Garkavi, The best possible net and the best possible cross-section of a set in a normed space. Izv. Akad. Nauk SSSR 26 (1962), 87–106. (Russian) (Translated in Amer. Math. Soc. Trans., Ser. 2, 39 (1964).)MathSciNetGoogle Scholar
- 7).M. Golomb and H. Weinberger, Optimal approximation and error bounds. On Numerical Approximation, R. Langer, Ed., Univ. of Wisconsin Press, Madison, 1959; 117–190.Google Scholar
- 8).A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type C(K). Can. J. Math. 5 (1953), 129–173.MathSciNetCrossRefMATHGoogle Scholar
- 9).N. Gurarii and Ju. Sozonov, Normed spaces in which the unit sphere has no bias. Math. Zametki 7 (1970), 307–310. (Russian) (Translated in Math. Notes 7 (1970), 187–189.)MathSciNetMATHGoogle Scholar
- 10).R. James and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction. Preprint.Google Scholar
- 11).M. Kadets and V. Zamyatin, Chebyshev centers in the space C[a,b]. Teo. Funk., Funkcion. Anal. Pril. 7 (1968), 20–26. (Russian).MATHGoogle Scholar
- 12).J. Kelley, Banach spaces with the extension property. Trans. Amer. Math. Soc. 72 (1952), 323–326.MathSciNetCrossRefMATHGoogle Scholar
- 13).P. Laurent and P. Dinh-Tuan, Global approximation of a compact set by elements of a convex set in a normed space. Num. Math. 15 (1970), 137–150.MathSciNetCrossRefMATHGoogle Scholar
- 14).J. Meinguet, Optimal approximation and interpolation in normed spaces. Numerical Approximation to Functions and Data, J. Hayes, Ed., Athlone Press, London, 1970; 143–157.Google Scholar
- 15).L. Nachbin, A theorem of the Hahn-Banach type for linear transformations. Trans. Amer. Math. Soc. 68 (1950), 28–46.MathSciNetCrossRefMATHGoogle Scholar
- 16).N. Routledge, A result in Hilbert space. Quart. J. Math. 3 (1952), 12–18.MathSciNetCrossRefMATHGoogle Scholar
- 17).M. Stone, Boundedness properties in function-lattices. Can. J. Math. 1 (1949), 176–186.MathSciNetCrossRefMATHGoogle Scholar
- 18).M. Valadier, Sous-différentiels d’une borne supérieure et d’une somme continue de fonctions convexes. C. R. Acad. Sci. Paris 268 (1969), A39–A42.MathSciNetMATHGoogle Scholar
References for §34
- 1).V. Ivanov, On linear problems which are not well posed. Soviet Math. Dokl. 3 (1962), 981–983.Google Scholar
- 2).M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics. Springer-Verlag, New York, 1967.CrossRefMATHGoogle Scholar
- 3).V. Tanana, Incorrectly posed problems and the geometry of Banach spaces. Soviet Math. Dokl. 11 (1970), 864–867.MATHGoogle Scholar
- 4).A. Tikhonov, On the stability of inverse problems. Dokl. Akad. Nauk SSSR 39 (1944), 195–198. (Russian)MathSciNetGoogle Scholar
References for §35
- 1).A. Ben-Israel, On iterative methods for solving nonlinear least squares problems over convex sets. Israel Math. J. 5 (1967), 211–224.MathSciNetCrossRefMATHGoogle Scholar
- 2).—, On Newton’s method in nonlinear programming, p. 339–352 in Princeton Symposium on Mathematical Programming (H. Kuhn, Ed.), Princeton Univ. Press, Princeton, 1970.Google Scholar
- 3).—, and A. Charnes, Contributions to the theory of generalized inverses. J. Soc. Ind. Appl. Math. 11 (1963), 667–699.MathSciNetCrossRefMATHGoogle Scholar
- 4).T. Boullion and P. Odell, Ed’s., Symposium on Theory and Application of Generalized Inverses of Matrices. Texas Tech. College, Lubbock, 1968.Google Scholar
- 5).—, Generalized Inverse Matrices. Wiley-Interscience, New York, 1971.MATHGoogle Scholar
- 6).H. Decell, An application of the Cayley-Hamilton Theorem to generalized matrix inversion. SIAM Rev. 7 (1965), 526–528.MathSciNetCrossRefMATHGoogle Scholar
- 7).I. Erdelyi and A. Ben-Israel, Extremal solutions of linear equations and generalized inversion between Hilbert spaces. J. Math. Anal. Appl., to appear.Google Scholar
- 8).R. Fletcher, Generalized inverses for nonlinear equations and optimization. p. 75–86 in Numerical Methods for Nonlinear Algebraic Equations (P. Rabinowitz, Ed.), Gordon and Breach, New York, 1970.Google Scholar
- 9).T. Greville, Some applications of the pseudoinverse of a matrix. SIAM Rev. 2 (1960), 15–22.MathSciNetCrossRefMATHGoogle Scholar
- 10).N. Minimide and K. Nakamura, A restricted pseudoinverse and its application to constrained minima. SIAM J. App. Math. 19 (1970), 167–177.MathSciNetCrossRefMATHGoogle Scholar
- 11).W. 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.MathSciNetCrossRefMATHGoogle Scholar
- 12).C. Price, The matrix pseudoinverse and minimal variance estimates. SIAM Rev. 6 (1964), 115–120.MathSciNetCrossRefMATHGoogle Scholar
- 13).D. Showalter, Representation and computation of the pseudoinverse. Proc. Amer. Math. Soc. 18 (1967), 584–586.MathSciNetCrossRefMATHGoogle Scholar
- 14).— and A. Ben-Israel, Representation and computation of the generalized inverse of a bounded linear operator between Hilbert spaces. Appl. Math. Report No. 69–12, Northwestern Univ., 1969.Google Scholar
- 15).S. Zlobec, Explicit computation of the Moore-Penrose generalized inverse. SIAM Rev. 12 (1970), 132–134.MathSciNetCrossRefMATHGoogle Scholar
Copyright information
© Springer-Verlag 1972