《INTERPOLATION AND APPROXIMATION》

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CHAPTER ⅠINTRODUCTION1

1.1 Determinants1

1.2 Solution of Linear Systems of Equations2

1.3 Linear Vector Spaces3

1.4 The Hierarchy of Functions4

1.5 Functions Satisfying a Lipschitz Condition8

1.6 Differentiable Functions8

1.7 Infinitely Differentiable Functions11

1.8 Functions Analytic on the Line12

1.9 Functions Analytic in a Region12

1.10 Entire Functions15

1.11 Polynomials15

1.12 Linear Functionals and the Algebraic Conjugate Space16

1.13 Some Assorted Facts19

CHAPTER ⅡINTERPOLATION24

2.1 Polynomial Interpolation24

2.2 The General Problem of Finite Interpolation26

2.3 Systems Possessing the Interpolation Property27

2.4 Unisolvence31

2.5 Representation Theorems:The Lagrange Formula33

2.6 Representation Theorems:The Newton Formula39

2.7 Successive Differences50

CHAPTER ⅢREMAINDER THEORY56

3.1 The Cauchy Remainder for Polynomial Interpolation56

3.2 Convex Functions58

3.3 Best Real Error Estimates;The Tschebyscheff Polynomials60

3.4 Divided Differences and Mean Values64

3.5 Interpolation at Coincident Points66

3.6 Analytic Functions:Remainder for Polynomial Interpolation67

3.7 Peano’s Theorem and Its Consequences69

3.8 Interpolation in Linear Spaces;General Remainder Theorem75

CHAPTER ⅣCONVERGENCE THEOREMS FOR INTERPOLATORY PROCESSES78

4.1 Approximation by Means of Interpolation78

4.2 Triangular Interpolation Schemes79

4.3 A Convergence Theorem for Bounded Triangular Schemes81

4.4 Lemniscates and Interpolation83

CHAPTER VSOME PROBLEMS OF INFINITE INTERPOLATION95

5.1 Characteristics of Such Problems95

5.2 Guichard’s Theorem96

5.3 A Second Approach:Infinite Systems of Linear Equations in Infinitely Many Unknowns97

5.4 Applications of Pólya’s Theorem102

CHAPTER ⅥUNIFORM APPROXIMATION107

6.1 The Weierstrass Approximation Theorem107

6.2 The Bernstein Polynomials108

6.3 Simultaneous Approximation of Functions and Derivatives112

6.4 Approximation by Interpolation:Fejér’s Proof118

6.5 Simultaneous Interpolation and Approximation121

6.6 Generalizations of the Weierstrass Theorem122

CHAPTER ⅦBEST APPROXIMATION128

7.1 What is Best Approximation?128

7.2 Normed Linear Spaces129

7.3 Convex Sets134

7.4 The Fundamental Problem of Linear Approximation136

7.5 Uniqueness of Best Approximation140

7.6 Best Uniform (Tsehebyscheff) Approximation of Continuous Functions146

7.7 Best Approximation by Nonlinear Families152

CHAPTER ⅧLEAST SQUARE APPROXIMATION158

8.1 Inner Product Spaces158

8.2 Angle Geometry for Inner Product Spaces161

8.3 Orthonormal Systems163

8.4 Fourier (or Orthogonal) Expansions169

8.5 Minimum Properties of Fourier Expansions171

8.6 The Normal Equations175

8.7 Gram Matrices and Determinants176

8.8 Further Properties of the Gram Determinant184

8.9 Closure and Its Consequences188

8.10 Further Geometrical Properties of Complete Inner Product Spaces195

CHAPTER ⅨHILBERT SPACE201

9.1 Introduction201

9.2 Three Hilbert Spaces203

9.3 Bounded Linear Functionals in Normed Linear Spaces and in Hilbert Spaces214

9.4 Linear Varieties and Hyperplanes;Interpolation and Approximation in Hilbert Space225

CHAPTER ⅩORTHOGONAL POLYNOMIALS234

10.1 General Properties of Real Orthogonal Polynomials234

10.2 Complex Orthogonal Polynomials239

10.3 The Special Function Theory of the Jacobi Polynomials246

CHAPTER ⅪTHE THEORY OF CLOSURE AND COMPLETENESS257

11.1 The Fundamental Theorem of Closure and Completeness257

11.2 Completeness of the Powers and Trigonometric Systems for L2[a,b]265

11.3 The Müntz Closure Theorem267

11.4 Closure Theorems for Classes of Analytic Functions273

11.5 Closure Theorems for Normed Linear Spaces281

CHAPTER ⅫEXPANSION THEOREMS FOR ORTHOGONAL FUNCTIONS290

12.1 The Historical Fourier Series290

12.2 Fejér’s Theory of Fourier Series299

12.3 Fourier Series of Periodic Analytic Functions305

12.4 Convergence of the Legendre Series for Analytic Functions308

12.5 Complex Orthogonal Expansions314

12.6 Reproducing Kernel Functions316

CHAPTER ⅩⅢDEGREE OF APPROXIMATION328

13.1 The Measure of Best Approximation328

13.2 Degree of Approximation with Some Square Norms333

13.3 Degree of Approximation with Uniform Norm334

CHAPTER ⅩⅣAPPROXIMATION OF LINEAR FUNCTIONALS341

14.1 Rules and Their Determination341

14.2 The Gauss-Jacobi Theory of Approximate Integration342

14.3 Norms of Functionals as Error Estimates345

14.4 Weak Convergence346

APPENDIX363

Short Guide to Orthogonal Polynomials365

Table of the Tschebyscheff Polynomials369

Table of the Legendre Polynomials371

BIBLIOGRAPHY373