Classical and modern Fourier analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Upper Saddle River, NJ
Pearson Education
2004
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Ab 2. Aufl. in 2 Bd. u.d.T.: "Classical Fourier analysis" bzw. "Modern Fourier analysis" |
| Physical Description: | Getr. Zählung Ill. |
| ISBN: | 013035399X |
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MARC
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|---|---|
| adam_text | CONTENTS
Preface ix
Chapter 1. IP Spaces and Interpolation 1
1.1. W and Weak IP 1
1.2. Convolution and Approximate Identities 16
1.3. Interpolation 32
1.4. Lorentz Spaces* 45
Chapter 2. Maximal Functions, Fourier Transform, and Distributions 77
2.1. Maximal Functions 78
2.2. The Schwartz Class and the Fourier Transform 94
2.3. The Class of Tempered Distributions 108
2.4. More about Distributions and the Fourier Transform* 122
2.5. Convolution Operators on Lp Spaces and Multipliers 133
2.6. Oscillatory Integrals 146
Chapter 3. Fourier Analysis on the Torus 157
3.1. Fourier Coefficients 157
3.2. Decay of Fourier Coefficients 172
3.3. Pointwise Convergence of Fourier Series 183
3.4. Divergence of Fourier Series and Bochner Riesz Summability* 192
3.5. The Conjugate Function and Convergence in Norm 208
3.6. Multipliers, Transference, and Almost Everywhere Convergence* 217
3.7. Lacunary Series* 235
Chapter 4. Singular Integrals of Convolution Type 247
4.1. The Hilbert Transform and the Riesz Transforms 247
4.2. Homogeneous Singular Integrals and the Method of Rotations 264
4.3. The Calderon Zygmund Decomposition and Singular Integrals 284
4.4. Sufficient Conditions for IP Boundedness 299
4.5. Vector Valued Inequalities* 311
4.6. Vector Valued Singular Integrals 325
V
Chapter 5. Littlewood Paley Theory and Multipliers 337
5.1. Littlewood Paley Theory 337
5.2. Two Multiplier Theorems 355
5.3. Applications of Littlewood Paley Theory 368
5.4. The Haar System, Conditional Expectation, and Martingales* 379
5.5. The Spherical Maximal Function* 390
5.6. Wavelets 397
Chapter 6. Smoothness and Function Spaces 413
6.1. Riesz Potentials, Bessel Potentials, and Fractional Integrals 413
6.2. Sobolev Spaces 424
6.3. Lipschitz Spaces 436
6.4. Hardy Spaces* 447
6.5. Besov Lipschitz and Triebel Lizorkin Spaces* 477
6.6. Atomic Decomposition* 487
6.7. Singular Integrals on Function Spaces 503
Chapter 7. BMO and Carleson Measures 517
7.1. Functions of Bounded Mean Oscillation 517
7.2. Duality between H1 and BMO 530
7.3. Nontangential Maximal Functions and Carleson Measures 535
7.4. The Sharp Maximal Function 545
7.5. Commutators of Singular Integrals with BMO Functions* 557
Chapter 8. Singular Integrals of Nonconvolution Type 569
8.1. General Background and the Role of BMO 569
8.2. Consequences of L2 Boundedness 584
8.3. The r(l) Theorem 590
8.4. Paraproducts 608
8.5. An Almost Orthogonality Lemma and Applications 620
8.6. The Cauchy Integral of Calderon and the T(b) Theorem* 634
8.7. Square Roots of Elliptic Operators* 652
Chapter 9. Weighted Inequalities 675
9.1. The Ap Condition 675
9.2. Reverse Holder Inequality for Ap Weights and Consequences 685
9.3. The Aoo condition* 694
9.4. Weighted Norm Inequalities for Singular Integrals 702
9.5. Further Properties of Ap Weights* 715
Chapter 10. Boundedness and Convergence of Fourier Integrals 733
10.1. The Multiplier Problem for the Ball 734
10.2. Bochner Riesz Means and the Carleson Sjolin Theorem 745
10.3. Kakeya Maximal Operators 762
10.4. Fourier Transform Restriction and Bochner Riesz Means 780
10.5. Almost Everywhere Convergence of Fourier Integrals* 796
10.6. IP Boundedness of the Carleson Operator* 831
Appendix A. Gamma and Beta Functions A l
A.I. A Useful Formula A l
A.2. Definitions of T(z) and B(z,w) A l
A.3. Volume of the Unit Ball and Surface of the Unit Sphere A 2
A.4. A Useful Integral A 3
A.5. Meromorphic Extensions of B(z,w) and T(z) A 3
A.6. Asymptotics of F(x) as x — oo A 4
A.7. The Duplication Formula for the Gamma Function A 5
Appendix B. Bessel Functions A 7
B.I. Definition A 7
B.2. Some Basic Properties A 7
B.3. An Interesting Identity A 9
B.4. The Fourier Transform of Surface Measure on Sn~1 A 10
B.5. The Fourier Transform of a Radial Function on Rra A 11
B.6. Asymptotics of Bessel Functions A ll
Appendix C. Rademacher Functions A 15
C.I. Definition of the Rademacher Functions A 15
C.2. Khintchine s Inequalities A 16
C.3. Derivation of Khintchine s Inequalities A 16
C.4. Khintchine s Inequalities for Weak Type Spaces A 19
C.5. Extension to Several Variables A 19
Appendix D. Spherical Coordinates A 21
D.I. Spherical Coordinate Formula A 21
D.2. A useful change of variables formula A 21
D.3. Computation of an Integral over the Sphere A 22
D.4. The Computation of Another Integral over the Sphere A 23
D.5. Integration over a General Surface A 23
D.6. The Stereographic Projection A 24
Appendix E. Some Trigonometric Identities and Inequalities A 25
Appendix F. Summation by Parts A 27
Appendix G. Basic Functional Analysis A 29
Appendix H. The Minimax Lemma A 31
Appendix I. The Schur Lemma A 35
1.1. The Classical Schur Lemma A 35
1.2. Schur s Lemma for Positive Operators A 36
1.3. An Example A 38
Appendix J. The Whitney Decomposition of Open Sets in Rn A 41
Appendix K. Smoothness and Vanishing Moments A 43
K.I. The Case of No Cancellation A 43
K.2. The Case of Cancellation A 44
K.3. The Case of Three Factors A 44
Bibliography B 47
Index of Notation 1 70
Index 1 73
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T20:05:12Z |
| institution | BVB |
| isbn | 013035399X |
| language | English |
| lccn | 2003051280 |
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| spelling | Grafakos, Loukas 1962- Verfasser (DE-588)136654444 aut Classical and modern Fourier analysis Loukas Grafakos Upper Saddle River, NJ Pearson Education 2004 Getr. Zählung Ill. txt rdacontent n rdamedia nc rdacarrier Ab 2. Aufl. in 2 Bd. u.d.T.: "Classical Fourier analysis" bzw. "Modern Fourier analysis" Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013073731&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grafakos, Loukas 1962- Classical and modern Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4023453-8 |
| title | Classical and modern Fourier analysis |
| title_auth | Classical and modern Fourier analysis |
| title_exact_search | Classical and modern Fourier analysis |
| title_full | Classical and modern Fourier analysis Loukas Grafakos |
| title_fullStr | Classical and modern Fourier analysis Loukas Grafakos |
| title_full_unstemmed | Classical and modern Fourier analysis Loukas Grafakos |
| title_short | Classical and modern Fourier analysis |
| title_sort | classical and modern fourier analysis |
| topic | Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Harmonische Analyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013073731&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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