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Approximation Theory and Methods
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.
Contents:
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.
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