Methods of Numerical Approximation

Methods of Numerical Approximation is based on lectures delivered at the Summer School held in September 1965, at Oxford University. The book deals with the approximation of functions with one or more variables, through means of more elementary functions. It explains systems to approximate functions, such as trigonometric sums, rational functions, continued fractions, and spline functions. The book also discusses linear approximation including topics such as convergence of polynomial interpolation and the least-squares approximation. The text analyzes Bernstein polynomials, Weierstrass' theorem, and Lagrangian interpolation. The book also gives attention to the Chebyshev least-squares approximation, the Chebyshev series, and the determination of Chebyshev series, under general methods. These general methods are useful when the student wants to investigate practical methods for finding forms of approximations under various situations. One of the lectures concerns the general theory of linear approximation and the existence of a best approximation approach using different theorems. The book also discusses the theory and calculation of the best rational approximations as well as the optimal approximation of linear functionals. The text will prove helpful for students in advanced mathematics and calculus. It can be appreciated by statisticians and those working with numbers theory.

Editor's Preface

I. General

1. Introduction

2. Some Abstract Concepts and Definitions

II. Linear Approximation

3. Convergence of Polynomial Interpolation

4. Least-Squares Approximation. Orthogonal Polynomials

5. Chebyshev Least-Squares Approximation

6. Determination and Properties of Chebyshev Expansions

7. The General Theory of Linear Approximation

8. The Exchange Algorithm on a Discrete Point Set

9. Calculation of the Best Linear Approximation on a Continuum

10. The Rate of Convergence of Best Approximations

III. Rational Approximation

11. Continued Fractions

12. Interpolation by Rational Functions

13. Economization of Continued Fractions

14. The Pade Table

15. Applications of the QD and ε Algorithms

16. Theory and Calculation of Best Rational Approximations

17. Convergence of Rational Approximations

IV. Miscellaneous

18. Theory of General Non-linear Minimax Approximation

19. Spline Functions

20. Optimal Approximation of Linear Functionals

21. Optimal Approximation by Means of Spline Functions

22. An Introduction to ε-Entropy

23. Functions of Many Variables

24. Practical Considerations

References

Further References

Index