Interpolation and Approximation

Cover
INTERPOLATION AND APPROXIMATION
Copyright
     © 1963 by Philip J. Davis.
     © 1975 by Philip J. Davis.
     ISBN 0486624951
     LCCN 75-2568
Dedication
Preface to the Dover Edition
Foreword

Contents

CHAPTER I  Introduction
     1.1 Determinants
     1.2 Solution of Linear Systems of Equations
     1.3 Linear Vector Spaces
     1.4 The Hierarchy of Functions
     1.5 Functions Satisfying a Lipschitz Condition
     1.6 Differentiable Functions
     1.7 Infinitely Differentiable Functions
     1.8 Functions Analytic on the Line
     1.9 Functions Analytic in a Region
     1.10 Entire Functions
     1.11 Polynomials
     1.12 Linear Functionals and the Algebraic Conjugate Space.
     1.13 Some Assorted Facts
     NOTES ON CHAPTER I
     PROBLEMS

CHAPTER II  Interpolation
     2.1 Polynomial Interpolation.
     2.2 The General Problem of Finite Interpolation
     2.3 Systems Possessing the Interpolation Property
     2.4 Unisolvence
     2.5 Representation Theorems: The Lagrange Formula
     2.6 Representation Theorems : The Newton Formula
     2.7 Successive Differences
     NOTES ON CHAPTER II
     PROBLEMS

CHAPTER III  Remainder Theory
     3.1 The Cauchy Remainder for Polynomial Interpolation
     3.2 Convex Functions
     3.3 Best Real Error Estimates; The Tschebyscheff Polynomials.
     3.4 Divided Differences and Mean Values
     3.5 Interpolation at Coincident Points
     3.6 Analytic Functions : Remainder for Polynomial Interpolation
     3.7 Peano\'s Theorem and Its Consequences
     3.8 . Interpolation in Linear Spaces; General Remainder Theorem.
     NOTES ON CHAPTER III
     PROBLEMS

CHAPTER IV  Convergence Theorems for Interpolatory Processes
     4.1 Approximation by Means of Interpolation
     4.2 Triangular Interpolation Schemes.
     4.3 A Convergence Theorem for Bounded Triangular Schemes.
     4.4 Lemniscates and Interpolation
     NOTES ON CHAPTER IV
     PROBLEMS

CHAPTER V  Some Problems of Infinite Interpolation
     5.1 Characteristics of Such Problems
     5.2 Guichard\'s Theorem
     5.3 A Second Approach: Infinite Systems of Linear Equations in Infinitely Many Unknowns
     5.4 Applications of Polya\'s Theorem.
     NOTES ON CHAPTER V
     PROBLEMS

CHAPTER VI  Uniform Approximation
     6.1 The Weierstrass Approximation Theorem
     6.2 The Bernstein Polynomials
     6.3 Simultaneous Approximation of Functions and Derivatives
     6.4 Approximation by Interpolation: Fejer\'s Proof.
     6.6 Generalizations of the Weierstrass Theorem
     NOTES ON CHAPTER VI
     PROBLEMS

CHAPTER VII  Best Approximation
     7.1 What is Best Approximation?
     7.2 Normed Linear Spaces
     7.3 Convex Sets.
     7.4 The Fundamental Problem of Linear Approximation
     7.5 Uniqueness of Best Approximation
     7.6 Best Uniform (Tschebyscheff) Approximation of Continuous Functions
     7.7 Best Approximation by Nonlinear Families
     NOTES ON CHAPTER VII
     PROBLEMS

CHAPTER VIII  Least Square Approximation
     8.1 Inner Product Spaces.
     8.2 Angle Geometry for Inner Product Spaces
     8.3 Orthonormal Systems
     8.4 Fourier (or Orthogonal) Expansions
     8.6 The Normal Equations
     8.7 Gram Matrices and Determinants
     8.8 Further Properties of the Gram Determinant
     8.9 Closure and Its Consequences
     8.10 Further Geometrical Properties of Complete Inner Product Spaces.
     NOTES ON CHAPTER VIII
     PROBLEMS

CHAPTER IX  Hilbert Space
     9.1 Introduction
     9.2 Three Hilbert Spaces.
     9.3 Bounded Linear Functionals in Normed Linear Spaces and in Hilbert Spaces
     9.4 Linear Varieties and Hyperplanes; Interpolation and Approximation in Hilbert Space
     NOTES ON CHAPTER IX
     PROBLEMS

CHAPTER X  Orthogonal Polynomials
     10.1 General Properties of Real Orthogonal Polynomials
     10.2 Complex Orthogonal Polynomials
     10.3 The Special Function Theory of the Jacobi Polynomials
     NOTES ON CHAPTER X
     PROBLEMS

CHAPTER XI  The Theory of Closure and Completeness
     11.1 The Fundamental Theorem of Closure and Completeness
     11.2 Completeness of the Powers and Trigonometric Systems for L2[a, b]
     11.3 The Miintz Closure Theorem
     11.4 Closure Theorems for Classes of Analytic Functions
     11.5 Closure Theorems for Normed Linear Spaces
     NOTES ON CHAPTER XI
     PROBLEMS

CHAPTER XII  Expansion Theorems for Orthogonal Functions
     12.1 The Historical Fourier Series
     12.2 Fejer\'s Theory of Fourier Series
     12.3 Fourier Series of Periodic Analytic Functions
     12.4 Convergence of the Legendre Series for Analytic Functions.
     12.5 Complex Orthogonal Expansions
     12.6 Reproducing Kernel Functions
     NOTES ON CHAPTER XII
     PROBLEMS

CHAPTER XIII  Degree of Approximation
     13.1 The Measure of Best Approximation
     13.2 Degree of Approximation with Some Square Norms
     13.3 Degree of Approximation with Uniform Norm
     NOTES ON CHAPTER XIII
     PROBLEMS

CHAPTER XIV  Approximation of Linear Functionals
     14.1 Rules and Their Determination.
     14.2 The Gauss-Jacobi Theory of Approximate Integration
     14.3 Norms of Functionals as Error Estimates
     14.4 Weak* Convergence
     NOTES ON CHAPTER XIV
     PROBLEMS

Appendix
     Short Guide to the Orthogonal Polynomials
     Table of the Tschebyscheff Polynomials
     Table of Powers as Combinations of the Tschebyscheff Polynomials
     Table of the Legendre Polynomials
     Table of Powers as Linear Combinations of the Legendre Polynomials

Bibliography
Index