Approximation Theory In Tensor Product Spaces

This is the first ever truly introductory text to the theory of tensor products of Banach spaces. Coverage includes a full treatment of the Grothendieck theory of tensor norms, approximation property and the Radon-Nikodym Property, Bochner and Pettis integrals. Each chapter contains worked examples and a set of exercises, and two appendices offer material on summability in Banach spaces and properties of spaces of measures.

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.

1. Introductory Discussion of Interpolation 2. Linear Interpolation Operators 3. Optimization of the Lagrange Operator 4. Multivariate Polynomials 5. Moving the Nodes 6. Projections 7. Tensor Product Interpolation 8. The Boolean Algebra of Projections 9. The Newton Paradigm for Interpolation 10. The Lagrange Paradigm for Interpolation 11. Interpolation by Translates of a Single Function 12. Positive Definite Functions 13. Strictly Positive Definite Functions 14. Completely Monotone Functions 15. The Schoenberg Interpolation Theorem 16. The Micchelli Interpolation Theorem 17. Positive Definite Functions of Spheres 18. Approximation by Positive Definite Functions 19. Approximate Reconstruction of Functions and Tomography 20. Approximation by Convolution 21. The Good Kernels 22. Ridge Functions 23. Ridge Function Approximation via Convolutions 24. Density of Ridge Functions 25. Artificial Neural Networks 26. Chebyshev Centers 27. Optimal Reconstruction of Functions 28. Algorithmic Orthogonal Projections 29. Cardinal B-Splines and the Sinc Function 30. The Golomb-Weinberger Theory 31. Hilbert Function Spaces, Reproducing Kernels 32. Spherical Splines 33. Box Splines 34. Wavelets, Part I 35. Wavelets, Part II 36. Quasi-Interpolation Bibliography / Index