Fourier Analysis and Approximation (eBook)

Front Cover 1
Fourier Analysis and Approximation: One–Dimensional Theory 4
Copyright Page 5
Contents 12
Chapter 0. Preliminaries 18
0.1 Fundamentals on Lebesgue Integration 18
0.2 Convolutions on the Line Group 21
0.3 Further Sets of Functions and Sequences 23
0.4 Periodic Functions and Their Convolution 25
0.5 Functions of Bounded Variation on the Line Group 27
0.6 The Class BV2p 31
0.7 Normed Linear Spaces, Bounded Linear Operators 32
0.8 Bounded Linear Functionals, Riesz Representation Theorems 37
0.9 References 41
Part I: Approximation by Singular Integrals 42
Chapter 1. Singular Integrals of Periodic Functions 46
1.0 Introduction 46
1.1 Norm-Convergence and -Derivatives 47
1.2 Summation of Fourier Series 56
1.3 Test Sets for Norm-Convergence 71
1.4 Pointwise Convergence 78
1.5 Order of Approximation for Positive Singular Integrals 84
1.6 Further Direct Approximation Theorems, NikolskiI Constants 96
1.7 Simple Inverse Approximation Theorems 103
1.8 Notes and Remarks 106
Chapter 2. Theorems of Jackson and Bernsteln for Polynomials of Best Approximation and for Singular Integrals 111
2.0 Introduction 111
2.1 Polynomials of Best Approximation 112
2.2 Theorems of Jackson 114
2.3 Theorems of Bernstein 116
2.4 Various Applications 121
2.5 Approximation Theorem for Singular Integrals 126
2.6 Notes and Remarks 133
Chapter 3. Singular Integrals on the Line Group 136
3.0 Introduction 136
3.1 Norm-Convergence 137
3.2 Pointwise Convergence 149
3.3 Order of Approximation 153
3.4 Further Direct Approximation Theorems 159
3.5 Inverse Approximation Theorems 163
3.6 Shape Preserving Properties 167
3.7 Notes and Remarks 175
Part II: Fourier Transforms 180
Chapter 4. Finite Fourier Transforms 184
4.0 Introduction 184
4.1 L½p-Theory 184
4.2 Lp 2p-Theory, P > 1
4.3 Finite Fourier–Stieltjes Transforms 196
4.4 Notes and Remarks 202
Chapter 5. Fourier Transforms Associated with the Line Group 205
5.0 Introduction 205
5.1 L1-Theory 205
5.2 Lp-Theory, 1< p=
5.3 Fourier–Stieltjes Transforms 236
5.4 Notes and Remarks 244
Chapter 6. Representation Theorems 248
6.0 Introduction 248
6.1 Necessary and Sufficient Conditions 249
6.2 Theorems of Bochner 258
6.3 Sufficient Conditions 263
6.4 Applications to Singular Integrals 273
6.5 Multipliers 283
6.6 Notes and Remarks 290
Chapter 7. Fourier Transform Methods and Second-Order Partial Differential Equations 295
7.0 Introduction 295
7.1 Finite Fourier Transform Method 298
7.2 Fourier Transform Method in L1 311
7.3 Notes and Remarks 317
Part III: Hilbert Transforms 320
Chapter 8. Hilbert Transforms on the Real Line 322
8.0 Introduction 322
8.1 Existence of the Transform 324
8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms 333
8.3 Fourier Transforms of Hilbert Transforms 341
8.4 Notes and Remarks 348
Chapter 9. Hilbert Transforms of Periodic Functions 351
9.0 Introduction 351
9.1 Existence and Basic Properties 352
9.2 Conjugates of Singular Integrals 358
9.3 Fourier Transforms of Hilbert Transforms 15
9.4 Notes and Remarks 364
Part IV: Characterization of Certain Function Classes 372
Chapter 10. Characterization in the Integral Case 374
10.0 Introduction 374
10.1 Generalized Derivatives, Characterization of the Classes Wr×2p 375
10.2 Characterization of the Classes Vr×2p 383
10.3 Characterization of the Classes (V `)r×2p 388
10.4 Relative Completion 390
10.5 Generalized Derivatives in LP-Norm and Characterizations for 1=p=2 393
10.6 Generalized Derivatives in X(R)-Norm and Characterizations of Classes W×(R) and Vr×(R) 399
10.7 Notes and Remarks 406
Chapter 11. characterization in the Fractional Case 408
11.0 Introduction 408
11.1 Integrals of Fractional Order 410
11.2 Characterizations of the Classes W[Lp ¦v¦a],V[Lp
11.3 The Operators R(a)eOn Lp,1=P=2 426
11.4 The Operators R(a)e On ×2p 433
11.5 Integral Representations, Fractional Derivatives of Periodic Functions 436
11.6 Notes and Remarks 445
Part V: Saturation Theory 448
Chapter 12.Saturation for Singular Integrals on X2pand LP,1=P=2 450
12.0 Introduction 450
12.1 Saturation for Periodic Singular Integrals, Inverse Theorems 452
12.2 Favard Classes 457
12.3 Saturation in Lp,1=P=2 469
12.4 Applications to Various Singular Integrals 480
12.5 Saturation of Higher Order 488
12.6 Notes and Remarks 495
Chapter 13. Saturation on X(R) 500
13.0 Introduction 500
13.1 Saturation of Ds(f x
13.2 Applications to Approximation in Lp, 2 < p<
13.3 Comparison Theorems 510
13.4 Saturation on Banach Spaces 519
13.5 Notes and Remarks 524
List of Symbols 528
Tables of Fourier and Hilbert Transforms 532
Bibliography 538
Index 564
Pure and Applied Mathematics 571