American Mathematical Society Colloquium Publications Volume XX Interpolation and ApproximationPDF电子书下载

CHAPTER Ⅰ POSSIBILITY OF APPROXIMATION;ANALYTIC FUNCTIONS 1

1．1．Point sets:preliminary definitions 1

1．2．Function-theoretic considerations 4

1．3．An open set as the sum of regions 6

1．4．Expansion of an analytic function 10

1．5．A theorem on analytic extension 12

1．6．Approximation;choice of poles 14

1．7．Components of an analytic function 17

1．8．Methods of Appell and of Wolff 19

1．9．On the vanishing of analytic functions 22

1．10．Necessary conditions for approximation 23

CHAPTER Ⅱ POSSIBILITY OF APPROXIMATION,CONTINOED 27

2．1．Lindelof＇s first theorem 27

2．2．Lindelof＇s second theorem 29

2．3．Conformal mapping of variable regions 32

2．4．Approximation in a closed Jordan region 36

2．5．Applications,Jordan configurations 39

2．6．General forms of Cauchy＇s integral formula 42

2．7．Surface integrals as measures of approximation 44

2．8．Uniform approximation;further results 46

CHAPTER Ⅲ INTERPOLATION AND LEMNISCATES 49

3．1．Polynomials of interpolation 49

3．2．Sequences and series of interpolation 52

3．3．Lemniscates and the Jacobi series 54

3．4．An analogous series of interpolation 56

3．5．A more general series of interpolation 60

CHAPTER Ⅳ DEGREE OF CONVERGENCE OF POLYNOMIALS．OVERCONVERGENCE 65

4．1．Equipotential curves in conformal maps 65

4．2．Approximation of Jordan curves by a lemniscate 68

4．3．Approximation of modulus of the mapping function 71

4．4．Approximation of modulus of mapping function,continued 74

4．5．Degree of convergence．Sufficient conditions 75

4．6．Degree of convergence．Necessary conditions．Overconvergence 77

4．7．Maximal convergence 79

4．8．Exact regions of uniform convergence 83

4．9．Approximation on more general point sets (irregular case) 85

CHAPTER Ⅴ BEST APPROXIMATION BY POLYNOMIALS 89

5．1．Tchebycheff approximation 89

5．2．Approximation measured by a line integral 91

5．3．Approximation measured by a surface integral 95

5．4．Approximation measured by a line integral after conformal mapping of complement 98

5．5．Approximation measured by a line integral after conformal mapping of interior 100

5．6．Point sets with infinitely many components 102

5．7．Generality of weight functions 104

5．8．Approximation of functions not analytic on closed set considered 107

CHAPTER Ⅵ ORTHOGONALITY AND LEAST WQUARES 111

6．1．Orthogonal functions and least squares 111

6．2．Orthogonalization 113

6．3．Riesz-Fischer theory 116

6．4．Closure 120

6．5．Polynomial approximation to analytic functions 125

6．6．Asymptotic properties of coefficients 128

6．7．Regions of convergence 131

6．8．Polynomials orthogonal on several curves 133

6．9．Functions of the second kind 136

6．10．Functions of class H2 141

6．11．Polynomials in z and 1/z 143

6．12．An extremal problem,line integrals 146

6．13．An extremal problem,surface integrals 149

CHAPTER Ⅶ INTERPOLATION BY POLYNOMIALS 152

7．1．Interpolation in roots of unity 152

7．2．A sufficient condition for maximal convergence 154

7．3．A necessary condition for uniform convergence 159

7．4．Further conditions for maximal convergence 162

7．5．Uniform distribution of points 164

7．6．Interpolation in points uniformly distributed 167

7．7．Points of interpolation with extremal properties 170

7．8．Existence of polynomials converging maximally (Shen) 173

7．9．A synthesis of interpolation and Tchebycheff approximation 175

7．10．Least squares and interpolation in roots of unity 178

CHAPTER Ⅷ INTERPOLATION BY RATIONAL FUNCTIONS 184

8．1．Interpolation formulas 184

8．2．Sequences and series of interpolation 188

8．3．Duality:general theorems 193

8．4．Duality:illustrations 199

8．5．Duality and series of interpolation 203

8．6．Illustrations 206

8．7．Harmonic functions as generating functions 209

8．8．Harmonic functions as generating functions,continued 212

8．9．Geometric conditions on given points 218

8．10．Geometric conditions,continuation 221

CHAPTER Ⅸ APPROXIMATION BY RATIONAL FUNCTIONS 224

9．1．Least squares on the unit circle and interpolation 224

9．2．Unit circle．Convergence theorems 228

9．3．Unit circle．Other measures of approximation 231

9．4．Unit circle．Asymptotic conditions on poles 235

9．5．Applications 239

9．6．Poles with limit points on circumference 243

9．7．General point sets;degree of convergence 249

9．8．General point sets;best approximation 252

9．9．Extensions 257

9．10．General point sets;asymptotic conditions on poles 261

9．11．Operations with asymptotic conditions 265

9．12．Asymptotic conditions under conformal transformation 270

9．13．Further problems 276

CHAPTER Ⅹ INTERPOLATION AND FUNCTIONS ANALYTIC IN THE UNIT CIRCLE 281

10．1．The Blaschke product 281

10．2．Functions of modulus not greater than M 286

10．3．Functions of least maximum modulus 290

10．4．Convergence of minimizing sequences 293

10．5．Totality of interpolating functions 296

10．6．Conditions for uniqueness 300

10．7．Functions of class H2 304

CHAPTER Ⅺ APPROXIMATION WITH AUXILIARY CONDITIONS AND TO NON-ANALYTIC FUNCTIONS 310

11．1．Approximation with interpolation to given function 310

11．2．Interpolation to given function;degree of convergence 314

11．3．Extremal problems involving auxiliary conditions 318

11．4．Expansion of mapping function 322

11．5．Approximation on a rectifiable Jordan curve 328

11．6．Interpolation in roots of unity 333

11．7．Tchebycheff measure of approximation;extremal problems 335

11．8．Tchebycheff approximation by polynomials and rational functions 338

11．9．Approximation by non-vanishing functions 343

CHAPTER Ⅻ EXISTENCE AND UNIQUENESS OF RATIONAL FUNCTIONS OF BEST APPROXIMATION 348

12．1．Sequences of rational functions of given degree 348

12．2．Application to Tchebycheff approximation 351

12．3．Restriction of location of poles 353

12．4．The rational function of best approximation need not be unique 356

12．5．Integral measures of approximation 357

12．6．Approximation with auxiliary conditions 360

12．7．Uniqueness of approximating functions with preassigned poles 361

A1．Possibility of approximation of polynomials 367

A2．Approximation by polynomials．Continuity conditions 371

A3．Interpolation and approximation by bounded analytic functions 373

BIBLIOGRAPHY 379

INDEX 391