Fourier Analysis and Approximation

1 - Front Cover [Seite 1]
2 - Fourier Analysis and Approximation: One-Dimensional Theory [Seite 4]
3 - Copyright Page [Seite 5]
4 - Contents [Seite 12]
4.1 - Chapter 0. Preliminaries [Seite 18]
4.1.1 - 0.1 Fundamentals on Lebesgue Integration [Seite 18]
4.1.2 - 0.2 Convolutions on the Line Group [Seite 21]
4.1.3 - 0.3 Further Sets of Functions and Sequences [Seite 23]
4.1.4 - 0.4 Periodic Functions and Their Convolution [Seite 25]
4.1.5 - 0.5 Functions of Bounded Variation on the Line Group [Seite 27]
4.1.6 - 0.6 The Class BV2p [Seite 31]
4.1.7 - 0.7 Normed Linear Spaces, Bounded Linear Operators [Seite 32]
4.1.8 - 0.8 Bounded Linear Functionals, Riesz Representation Theorems [Seite 37]
4.1.9 - 0.9 References [Seite 41]
5 - Part I: Approximation by Singular Integrals [Seite 42]
5.1 - Chapter 1. Singular Integrals of Periodic Functions [Seite 46]
5.1.1 - 1.0 Introduction [Seite 46]
5.1.2 - 1.1 Norm-Convergence and -Derivatives [Seite 47]
5.1.3 - 1.2 Summation of Fourier Series [Seite 56]
5.1.4 - 1.3 Test Sets for Norm-Convergence [Seite 71]
5.1.5 - 1.4 Pointwise Convergence [Seite 78]
5.1.6 - 1.5 Order of Approximation for Positive Singular Integrals [Seite 84]
5.1.7 - 1.6 Further Direct Approximation Theorems, NikolskiI Constants [Seite 96]
5.1.8 - 1.7 Simple Inverse Approximation Theorems [Seite 103]
5.1.9 - 1.8 Notes and Remarks [Seite 106]
5.2 - Chapter 2. Theorems of Jackson and Bernsteln for Polynomials of Best Approximation and for Singular Integrals [Seite 111]
5.2.1 - 2.0 Introduction [Seite 111]
5.2.2 - 2.1 Polynomials of Best Approximation [Seite 112]
5.2.3 - 2.2 Theorems of Jackson [Seite 114]
5.2.4 - 2.3 Theorems of Bernstein [Seite 116]
5.2.5 - 2.4 Various Applications [Seite 121]
5.2.6 - 2.5 Approximation Theorem for Singular Integrals [Seite 126]
5.2.7 - 2.6 Notes and Remarks [Seite 133]
5.3 - Chapter 3. Singular Integrals on the Line Group [Seite 136]
5.3.1 - 3.0 Introduction [Seite 136]
5.3.2 - 3.1 Norm-Convergence [Seite 137]
5.3.3 - 3.2 Pointwise Convergence [Seite 149]
5.3.4 - 3.3 Order of Approximation [Seite 153]
5.3.5 - 3.4 Further Direct Approximation Theorems [Seite 159]
5.3.6 - 3.5 Inverse Approximation Theorems [Seite 163]
5.3.7 - 3.6 Shape Preserving Properties [Seite 167]
5.3.8 - 3.7 Notes and Remarks [Seite 175]
6 - Part II: Fourier Transforms [Seite 180]
6.1 - Chapter 4. Finite Fourier Transforms [Seite 184]
6.1.1 - 4.0 Introduction [Seite 184]
6.1.2 - 4.1 L½p-Theory [Seite 184]
6.1.3 - 4.2 Lp 2p-Theory, P > 1 [Seite 191]
6.1.4 - 4.3 Finite Fourier-Stieltjes Transforms [Seite 196]
6.1.5 - 4.4 Notes and Remarks [Seite 202]
6.2 - Chapter 5. Fourier Transforms Associated with the Line Group [Seite 205]
6.2.1 - 5.0 Introduction [Seite 205]
6.2.2 - 5.1 L1-Theory [Seite 205]
6.2.3 - 5.2 Lp-Theory, 1

6.2.4 - 5.3 Fourier-Stieltjes Transforms [Seite 236]
6.2.5 - 5.4 Notes and Remarks [Seite 244]
6.3 - Chapter 6. Representation Theorems [Seite 248]
6.3.1 - 6.0 Introduction [Seite 248]
6.3.2 - 6.1 Necessary and Sufficient Conditions [Seite 249]
6.3.3 - 6.2 Theorems of Bochner [Seite 258]
6.3.4 - 6.3 Sufficient Conditions [Seite 263]
6.3.5 - 6.4 Applications to Singular Integrals [Seite 273]
6.3.6 - 6.5 Multipliers [Seite 283]
6.3.7 - 6.6 Notes and Remarks [Seite 290]
6.4 - Chapter 7. Fourier Transform Methods and Second-Order Partial Differential Equations [Seite 295]
6.4.1 - 7.0 Introduction [Seite 295]
6.4.2 - 7.1 Finite Fourier Transform Method [Seite 298]
6.4.3 - 7.2 Fourier Transform Method in L1 [Seite 311]
6.4.4 - 7.3 Notes and Remarks [Seite 317]
7 - Part III: Hilbert Transforms [Seite 320]
7.1 - Chapter 8. Hilbert Transforms on the Real Line [Seite 322]
7.1.1 - 8.0 Introduction [Seite 322]
7.1.2 - 8.1 Existence of the Transform [Seite 324]
7.1.3 - 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms [Seite 333]
7.1.4 - 8.3 Fourier Transforms of Hilbert Transforms [Seite 341]
7.1.5 - 8.4 Notes and Remarks [Seite 348]
7.2 - Chapter 9. Hilbert Transforms of Periodic Functions [Seite 351]
7.2.1 - 9.0 Introduction [Seite 351]
7.2.2 - 9.1 Existence and Basic Properties [Seite 352]
7.2.3 - 9.2 Conjugates of Singular Integrals [Seite 358]
7.2.4 - 9.3 Fourier Transforms of Hilbert Transforms [Seite 15]
7.2.5 - 9.4 Notes and Remarks [Seite 364]
8 - Part IV: Characterization of Certain Function Classes [Seite 372]
8.1 - Chapter 10. Characterization in the Integral Case [Seite 374]
8.1.1 - 10.0 Introduction [Seite 374]
8.1.2 - 10.1 Generalized Derivatives, Characterization of the Classes Wr×2p [Seite 375]
8.1.3 - 10.2 Characterization of the Classes Vr×2p [Seite 383]
8.1.4 - 10.3 Characterization of the Classes (V `)r×2p [Seite 388]
8.1.5 - 10.4 Relative Completion [Seite 390]
8.1.6 - 10.5 Generalized Derivatives in LP-Norm and Characterizations for 1=p=2 [Seite 393]
8.1.7 - 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of Classes W×(R) and Vr×(R) [Seite 399]
8.1.8 - 10.7 Notes and Remarks [Seite 406]
8.2 - Chapter 11. characterization in the Fractional Case [Seite 408]
8.2.1 - 11.0 Introduction [Seite 408]
8.2.2 - 11.1 Integrals of Fractional Order [Seite 410]
8.2.3 - 11.2 Characterizations of the Classes W[Lp [Seite 417]
8.2.4 - 11.3 The Operators R(a)eOn Lp,1=P=2 [Seite 426]
8.2.5 - 11.4 The Operators R(a)e On ×2p [Seite 433]
8.2.6 - 11.5 Integral Representations, Fractional Derivatives of Periodic Functions [Seite 436]
8.2.7 - 11.6 Notes and Remarks [Seite 445]
9 - Part V: Saturation Theory [Seite 448]
9.1 - Chapter 12.Saturation for Singular Integrals on X2pand LP,1=P=2 [Seite 450]
9.1.1 - 12.0 Introduction [Seite 450]
9.1.2 - 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems [Seite 452]
9.1.3 - 12.2 Favard Classes [Seite 457]
9.1.4 - 12.3 Saturation in Lp,1=P=2 [Seite 469]
9.1.5 - 12.4 Applications to Various Singular Integrals [Seite 480]
9.1.6 - 12.5 Saturation of Higher Order [Seite 488]
9.1.7 - 12.6 Notes and Remarks [Seite 495]
9.2 - Chapter 13. Saturation on X(R) [Seite 500]
9.2.1 - 13.0 Introduction [Seite 500]
9.2.2 - 13.1 Saturation of Ds(f [Seite 502]
9.2.3 - 13.2 Applications to Approximation in Lp, 2 9.2.4 - 13.3 Comparison Theorems [Seite 510]
9.2.5 - 13.4 Saturation on Banach Spaces [Seite 519]
9.2.6 - 13.5 Notes and Remarks [Seite 524]
10 - List of Symbols [Seite 528]
11 - Tables of Fourier and Hilbert Transforms [Seite 532]
12 - Bibliography [Seite 538]
13 - Index [Seite 564]
14 - Pure and Applied Mathematics [Seite 571]