Interpolation and approximation offer important applications in computer science and elsewhere. This intermediate-level survey abounds in useful examples of related subjects, starting with remainder theory, convergence theorems, and uniform and best approximation. Other topics include least square approximation, Hilbert space, orthogonal polynomials, the theory of closure and completeness, and more. 1963 edition.
Author(s): Philip J. Davis
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 1975
Language: English
Pages: xvi+393
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
