三角级数 第2卷 第3版PDF电子书下载

CHAPTER Ⅹ TRIGONOMETRIC INTERPOLATION 1

1．General remarks 1

2．Interpolating polynomials as Fourier series 6

3．The case of an even number of fundamental points 8

4．Fourier-Lagrange coefficients 14

5．Convergence of interpolating polynomials 16

6．Jackson polynomials and related topics 21

7．Mean convergence of interpolating polynomials 27

8．Divergence ofinterpolating polynomials 35

9．Divergence of interpolating polynomials（cont.） 44

10．Polynomials conjugate to interpolating polynomials 48

Miscellaneous theorems and examples 55

CHAPTER Ⅺ DIFFERENTIATION OF SERIES.GENERALIZED DERIVATIVES 55

1．Cesàro summability of differentiated series 59

2．Summability C of Fourier series 65

3．A theorem on differentiated series 71

4．Theorems on generalized derivatives 73

5．Applications of Theorem（4·2）to Fourier series 80

6．The integral M and Fourier series 83

7．The integral M2 86

Miscellaneous theorems and examples 91

CHAPTER Ⅻ INTERPOLATION OF LINEAR OPERATIONS.MORE ABOUT FOURIER COEFFICIENTS 91

1．The Riesz-Thorin theorem 93

2．The theorems of Hausdorff-Young and F.Riesz 101

3．Interpolation of operations in the classes Hr 105

4．Marcinkiewicz's theorem on the interpolation of operations 111

5．Paley's theorems on Fourier coefficients 120

6．Theorems of Hardy and Littlewood about rearrangements of Fourier coefficients 127

7．Lacunary coefficients 131

8．Fractional integration 133

9．Fractional integration（cont.） 138

10．Fourier-Stieltjes coefficients 142

11．Fourier-Stieltjes coefficients and sets of constant ratio of dissection 147

Miscellaneous theorems and examples 156

CHAPTER ⅩⅢ CONVERGENCE AND SUMMABILITY ALMOST EVERYWHERE 156

1．Partial sums of S［f］for f∈L2 161

2．Order of magnitude of Sn for f∈Lp 166

3．A test for the convergence of S［f］almost everywhere 170

4．Majorants for the partial sums of S［f］and ？［f］ 173

5．Behaviour of the partial sums of S［f］and ？［f］ 175

6．Theorems on the partial sums of power sories 178

7．Strong summability of Fourier series.The case f∈Lr,r＞1 180

8．Strong summability of S［f］and ？［f］in the general case 184

9．Almost convergence of S［f］and ？［f］ 188

10．Theorems on the convergence of orthogonal series 189

11．Capacity of sets and convergence of Fourier series 194

Miscellaneous theorems and examples 197

CHAPTER ⅩⅣ MORE ABOUT COMPLEX METHODS 199

1．Boundary behaviour of harmonic and analytic functions 199

2．The function s（θ） 207

3．The Littlewood-Paley function g（θ） 210

4．Convergence of conjugate series 216

5．The Marcinkiewicz functionμ（θ） 219

Miscellaneous theorems and examples 221

CHAPTER ⅩⅤ APPLICATIONS OF THE LITTLEWOOD-PALEY FUNCTION TO FOURIER SERIES 221

1．General remarks 222

2．Functions in Lr,l＜r＜∞ 224

3．Functions in Lr,l＜r＜∞（cont.） 229

4．Theorems on the partial sums of S［f］,f∈Lr,l＜r＜∞ 230

5．The limiting case r=l 234

6．The limiting case r=∞ 239

CHAPTER ⅩⅥ FOURIER INTEGRALS 242

1．General remarks 242

2．Fourier transforms 246

3．Fourier transforms（cont.） 254

4．Fourier-Stieltjes transforms 258

5．Applications to trigonometric series 263

6．Applications to trigonometric series（cont.） 269

7．The Paley-Wiener theorem 272

8．Riemann theory of trigonometric integrals 278

9．Equiconvergence theorems 286

10．Problems of uniqueness 291

Miscellaneous theorems and examples 297

CHAPTER ⅩⅦ A TOPIC IN MULTIPLE FOURIER SERIES 300

1．General remarks 300

2．Strong differentiability of multiple integrals and its applications 305

3．Restricted summability of Fourier series 309

4．Power series of several variables 315

5．Power series of several variables（cont.） 321

Miscellaneous theorems and examples 328

Notes 331

Bibtiography 336

Index 353