Approximation Theory and Methods

M. J. D. Powell (Author)

9780521295147, Cambridge University Press

Paperback, published 31 March 1981

352 pages
23.1 x 15.4 x 2.2 cm, 0.487 kg

"Of its kind, this book is excellent, perhaps the best." Journal of Classification

Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.

Preface
1. The approximation problem and existence of best approximations
2. The uniqueness of best approximations
3. Approximation operators and some approximating functions
4. Polynomial interpolation
5. Divided differences
6. The uniform convergence of polynomial approximations
7. The theory of minimax approximation
8. The exchange algorithm
9. The convergence of the exchange algorithm
10. Rational approximation by the exchange algorithm
11. Least squares approximation
12. Properties of orthogonal polynomials
13. Approximation of periodic functions
14. The theory of best L1 approximation
15. An example of L1 approximation and the discrete case
16. The order of convergence of polynomial approximations
17. The uniform boundedness theorem
18. Interpolation by piecewise polynomials
19. B-splines
20. Convergence properties of spline approximations
21. Knot positions and the calculation of spline approximations
22. The Peano kernel theorem
23. Natural and perfect splines
24. Optimal interpolation
Appendices
Index.

Subject Areas: Numerical analysis [PBKS]