Author | Deutsch, Frank. author |
---|---|
Title | Best Approximation in Inner Product Spaces [electronic resource] / by Frank Deutsch |
Imprint | New York, NY : Springer New York, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9298-9 |
Descript | XVI, 338 p. online resource |
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 -- Mรผntzโs 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 Alternating Projections -- The Case of Two Subspaces -- Angle Between Two Subspaces -- Rate of Convergence for Alternating Projections (two subspaces) -- Weak Convergence -- Dykstraโs 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. Zornโs Lemma -- References