Methods of Numerical Approximation

Lectures Delivered at a Summer School Held at Oxford University, September 1965

D. C. Handscomb

• Publisher: Pergamon
• Year: 2014
• Print ISBN:
  9780080119960
• eBook ISBN:
  9781483149028
• Topic: Social Sciences
• Subtopic: Anthropology
• Index: Social Sciences

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.

Table of contents

• Front Cover
• Methods of Numerical Approximation
• Copyright Page
• Table of Contents
• EDITOR'S PREFACE
• PART I: GENERAL
• CHAPTER 1. INTRODUCTION
• CHAPTER 2. SOME ABSTRACT CONCEPTS ANDDEFINITIONS
• PART II: LINEAR APPROXIMATION
• CHAPTER 3. CONVERGENCE OF POLYNOMIAL INTERPOLATION
• CHAPTER 4. LEAST-SQUARES APPROXIMATION. ORTHOGONAL POLYNOMIALS
• CHAPTER 5. CHEBYSHEV LEAST-SQUARES APPROXIMATION
• 1. THE FOURIER SERIES. CONVERGENCE IN THE MEAN
• 2. THE FOURIER SERIES. POINT-WISE CONVERGENCE
• 3. THE CHEBYSHEV SERIES
• 4. THE CHEBYSHEV POLYNOMIALS. TWO DISCRETE LEAST-SQUARES SOLUTIONS
• 5. RELATIVE CONVERGENCE OF CHEBYSHEV AND OTHER SERIES
• CHAPTER 6. DETERMINATION AND PROPERTIES OF CHEBYSHEV EXPANSIONS
• CHAPTER 7. THE GENERAL THEORY OF LINEAR APPROXIMATION
• CHAPTER 8. THE EXCHANGE ALGORITHM FOR CALCULATING MINIMAX LINEAR APPROXIMATIONS OVER A DISCRETE POINT SET
• CHAPTER 9. CALCULATION OF THE BEST LINEAR APPROXIMATION ON A CONTINUUM
• CHAPTER 10. THE RATE OF CONVERGENCE OF BEST APPROXIMATIONS
• PART III: RATIONAL APPROXIMATION
• CHAPTER 11. CONTINUED FRACTIONS
• CHAPTER 12. INTERPOLATION BY RATIONAL FUNCTIONS
• CHAPTER 13. ECONOMIZATION OF CONTINUED FRACTIONS
• CHAPTER 14. THE PADÉ TABLE
• CHAPTER 15. APPLICATIONS OF THE QD AND ε ALGORITHMS
• CHAPTER 16. THEORY AND CALCULATION OF BEST RATIONAL APPROXIMATIONS
• CHAPTER 17. CONVERGENCE OF RATIONAL APPROXIMATIONS
• PART IV: MISCELLANEOUS
• CHAPTER 18. THEORY OF GENERAL NON-LINEAR MINIMAX APPROXIMATIONS
• CHAPTER 19. SPLINE FUNCTIONS
• CHAPTER 20. OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS
• CHAPTER 21. OPTIMAL APPROXIMATION BY MEANS OF SPLINE FUNCTIONS
• CHAPTER 22. AN INTRODUCTION TO ε-ENTROPY
• CHAPTER 23. FUNCTIONS OF MANY VARIABLES
• CHAPTER 24. PRACTICAL CONSIDERATIONS
• REFERENCES
• FURTHER REFERENCES
• INDEX