Best Approximation in Inner Product Spaces

1. Auflage. Softcover version of original hardcover Auflage 2001
ISBN: 978-1-4419-2890-0
Verlag: Springer

Buch, Englisch, 338 Seiten, Previously published in hardcover, Format (B × H): 155 mm x 235 mm, Gewicht: 540 g

Reihe: CMS Books in Mathematics

This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisites for reading the book is some knowledge of advanced calculus and linear algebra.

Zielgruppe

Graduate

Autoren/Hrsg.

Deutsch, Frank R.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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.