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＜電子ブック＞
Best Approximation in Inner Product Spaces / by Frank R. Deutsch
(CMS Books in Mathematics, Ouvrages de mathématiques de la SMC. ISSN:21974152)

版	1st ed. 2001.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	2001
本文言語	英語
大きさ	XVI, 338 p : online resource
著者標目	*Deutsch, Frank R author SpringerLink (Online service)
件　名	LCSH:Mathematical analysis LCSH:Computer science -- Mathematics  全ての件名で検索 FREE:Analysis FREE:Mathematics of Computing
一般注記	1. Inner Product Spaces -- Five Basic Problems -- Inner Product Spaces -- Orthogonality -- Topological Notions -- Hilbert Space -- Exercises -- Historical Notes -- 2. Best Approximation -- Best Approximation -- Convex Sets -- Five Basic Problems Revisited -- Exercises -- Historical Notes -- 3. Existence and Uniqueness of Best Approximations -- Existence of Best Approximations -- Uniqueness of Best Approximations -- Compactness Concepts -- Exercises -- Historical Notes -- 4. Characterization of Best Approximations -- Characterizing Best Approximations -- Dual Cones -- Characterizing Best Approximations from Subspaces -- Gram-Schmidt Orthonormalization -- Fourier Analysis -- Solutions to the First Three Basic Problems -- Exercises -- Historical Notes -- 5. The Metric Projection -- Metric Projections onto Convex Sets -- Linear Metric Projections -- The Reduction Principle -- Exercises -- Historical Notes -- 6. Bounded Linear Functionals and Best Approximation from Hyperplanes and Half-Spaces -- Bounded Linear Functionals -- Representation of Bounded Linear Functionals -- Best Approximation from Hyperplanes -- Strong Separation Theorem -- Best Approximation from Half-Spaces -- Best Approximation from Polyhedra -- Exercises -- Historical Notes -- 7. Error of Approximation -- Distance to Convex Sets -- Distance to Finite-Dimensional Subspaces -- Finite-Codimensional Subspaces -- The Weierstrass Approximation Theorem -- Müntz’s Theorem -- Exercises -- Historical Notes -- 8. Generalized Solutions of Linear Equations -- Linear Operator Equations -- The Uniform Boundedness and Open Mapping Theorems -- The Closed Range and Bounded Inverse Theorems -- The Closed Graph Theorem -- Adjoint of a Linear Operator -- Generalized Solutions to Operator Equations -- Generalized Inverse -- Exercises -- Historical Notes -- 9. The Method of AlternatingProjections -- The Case of Two Subspaces -- Angle Between Two Subspaces -- Rate of Convergence for Alternating Projections (two subspaces) -- Weak Convergence -- Dykstra’s Algorithm -- The Case of Affine Sets -- Rate of Convergence for Alternating Projections -- Examples -- Exercises -- Historical Notes -- 10. Constrained Interpolation from a Convex Set -- Shape-Preserving Interpolation -- Strong Conical Hull Intersection Property (Strong CHIP) -- Affine Sets -- Relative Interiors and a Separation Theorem -- Extremal Subsets of C -- Constrained Interpolation by Positive Functions -- Exercises -- Historical Notes -- 11. Interpolation and Approximation -- Interpolation -- Simultaneous Approximation and Interpolation -- Simultaneous Approximation, Interpolation, and Norm-preservation -- Exercises -- Historical Notes -- 12. Convexity of Chebyshev Sets -- Is Every Chebyshev Set Convex? -- Chebyshev Suns -- Convexity of Boundedly Compact Chebyshev Sets -- Exercises -- Historical Notes -- Appendix 1. Zorn’s Lemma -- References This book evolved from notes originally developed for a graduate course, "Best Approximation in Normed Linear Spaces," that I began giving at Penn State Uni­ versity more than 25 years ago. It soon became evident. that many of the students who wanted to take the course (including engineers, computer scientists, and statis­ ticians, as well as mathematicians) did not have the necessary prerequisites such as a working knowledge of Lp-spaces and some basic functional analysis. (Today such material is typically contained in the first-year graduate course in analysis. ) To accommodate these students, I usually ended up spending nearly half the course on these prerequisites, and the last half was devoted to the "best approximation" part. I did this a few times and determined that it was not satisfactory: Too much time was being spent on the presumed prerequisites. To be able to devote most of the course to "best approximation," I decided to concentrate on the simplest of the normed linear spaces-the inner product spaces-since the theory in inner product spaces can be taught from first principles in much less time, and also since one can give a convincing argument that inner product spaces are the most important of all the normed linear spaces anyway. The success of this approach turned out to be even better than I had originally anticipated: One can develop a fairly complete theory of best approximation in inner product spaces from first principles, and such was my purpose in writing this book HTTP:URL=https://doi.org/10.1007/978-1-4684-9298-9

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