Methods of Numerical Approximation: Lectures Delivered At a Summer School Held At Oxford University, September, 1965.

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. Content: Front Matter, Page iii Copyright, Page iv Dedication, Page v EDITOR'S PREFACE, Page ix, D.C. HANDSCOMB CHAPTER 1 - INTRODUCTION, Pages 3-5, D.C. HANDSCOMB CHAPTER 2 - SOME ABSTRACT CONCEPTS AND DEFINITIONS, Pages 7-12, D.C. HANDSCOMB CHAPTER 3 - CONVERGENCE OF POLYNOMIAL INTERPOLATION, Pages 15-26, D.F. MAYERS CHAPTER 4 - LEAST-SQUARES APPROXIMATION. ORTHOGONAL POLYNOMIALS, Pages 27-37, L. FOX CHAPTER 5 - CHEBYSHEV LEAST-SQUARES APPROXIMATION, Pages 39-45, L. FOX CHAPTER 6 - DETERMINATION AND PROPERTIES OF CHEBYSHEV EXPANSIONS, Pages 47-59, L. FOX CHAPTER 7 - THE GENERAL THEORY OF LINEAR APPROXIMATION, Pages 61-71, D.C. HANDSCOMB, D.F. MAYERS, M.J.D. POWELL CHAPTER 8 - THE EXCHANGE ALGORITHM FOR CALCULATING MINIMAX LINEAR APPROXIMATIONS OVER A DISCRETE POINT SET, Pages 73-81, M.J.D. POWELL CHAPTER 9 - CALCULATION OF THE BEST LINEAR APPROXIMATION ON A CONTINUUM, Pages 83-89, A.R. CURTIS CHAPTER 10 - THE RATE OF CONVERGENCE OF BEST APPROXIMATIONS, Pages 91-96, D.C. HANDSCOMB CHAPTER 11 - CONTINUED FRACTIONS, Pages 99-103, J.D.P. DONNELLY CHAPTER 12 - INTERPOLATION BY RATIONAL FUNCTIONS, Pages 105-116, D.F. MAYERS CHAPTER 13 - ECONOMIZATION OF CONTINUED FRACTIONS, Pages 117-123, D.F. MAYERS CHAPTER 14 - THE PADÉ TABLE, Pages 125-130, J.D.P. DONNELLY CHAPTER 15 - APPLICATIONS OF THE QD AND ɛ ALGORITHMS, Pages 131-138, J.D.P. DONNELLY CHAPTER 16 - THEORY AND CALCULATION OF BEST RATIONAL APPROXIMATIONS, Pages 139-148, A.R. CURTIS CHAPTER 17 - CONVERGENCE OF RATIONAL APPROXIMATIONS, Pages 149-152, D.C. HANDSCOMB CHAPTER 18 - THEORY OF GENERAL NON-LINEAR MINIMAX APPROXIMATIONS, Pages 155-162, M.J.D. POWELL CHAPTER 19 - SPLINE FUNCTIONS, Pages 163-167, D.C. HANDSCOMB CHAPTER 20 - OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS, Pages 169-176, D.C. HANDSCOMB CHAPTER 21 - OPTIMAL APPROXIMATION BY MEANS OF SPLINE FUNCTIONS, Pages 177-181, D.C. HANDSCOMB CHAPTER 22 - AN INTRODUCTION TO ε-ENTROPY, Pages 183-190, M.J.D. POWELL CHAPTER 23 - FUNCTIONS OF MANY VARIABLES, Pages 191-194, D.C. HANDSCOMB CHAPTER 24 - PRACTICAL CONSIDERATIONS, Pages 195-198, D.C. HANDSCOMB REFERENCES, Pages 199-202 FURTHER REFERENCES, Pages 203-214 INDEX, Pages 215-218