Gespeichert in:
| Titel: | Classical Fourier analysis |
|---|---|
| Von: |
Loukas Grafakos
|
| Person: |
Grafakos, Loukas
1962- Verfasser aut |
| Vorheriger Titel: | Grafakos, Loukas Classical and modern Fourier analysis |
| Hauptverfasser: | |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York, NY
Springer
2008
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
249 |
| Schlagworte: | |
| Online-Zugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754122&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke, 1. Aufl. (in 1 Bd. zs. mit 1. Aufl. von Grafakos, Loukas: Modern Fourier analysis) u.d.T.: Grafakos, Loukas: Classical and modern Fourier analysis |
| Beschreibung: | XVI, 489 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9780387094311 9780387094328 9781441918550 |
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Datensatz im Suchindex
| DE-BY-OTHR_call_number | F 03/SK 450 G736 C6(2) |
|---|---|
| DE-BY-OTHR_katkey | 5081836 |
| DE-BY-OTHR_location | 02 |
| DE-BY-OTHR_media_number | 067004609626 |
| DE-BY-TUM_call_number | 0102 MAT 420f 2009 A 4850(2) |
| DE-BY-TUM_katkey | 1695980 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010153376 |
| DE-BY-UBR_call_number | 80/SK 450 G736 C6(2) 8014/SK 450 G736 C6(2) |
| DE-BY-UBR_katkey | 5081836 |
| DE-BY-UBR_location | UB Lesesaal Mathematik UB Handapparat Mathematik Prof. Abels |
| DE-BY-UBR_media_number | 069037298631 TEMP12434621 |
| _version_ | 1835111114574135296 |
| adam_text | Contents
1 LP Spaces and Interpolation 1
1.1 LP and Weak LP 1
1.1.1 The Distribution Function 2
1.1.2 Convergence in Measure 5
1.1.3 A First Glimpse at Interpolation 8
Exercises 10
1.2 Convolution and Approximate Identities 16
1.2.1 Examples of Topological Groups 16
1.2.2 Convolution 18
1.2.3 Basic Convolution Inequalities 19
1.2.4 Approximate Identities 24
Exercises 28
1.3 Interpolation 30
1.3.1 Real Method: The Marcinkiewicz Interpolation Theorem ... 31
1.3.2 Complex Method: The Riesz-Thorin Interpolation Theorem.. 34
1.3.3 Interpolation of Analytic Families of Operators 37
1.3.4 Proofs of Lemmas 1.3.5 and 1.3.8 39
Exercises 42
1.4 Lorentz Spaces 44
1.4.1 Decreasing Rearrangements 44
1.4.2 Lorentz Spaces 48
1.4.3 Duals of Lorentz Spaces 51
1.4.4 The Off-Diagonal Marcinkiewicz Interpolation Theorem ... 55
Exercises 63
2 Maximal Functions, Fourier Transform, and Distributions 77
2.1 Maximal Functions 78
2.1.1 The Hardy-Littlewood Maximal Operator 78
2.1.2 Control of Other Maximal Operators 82
2.1.3 Applications to Differentiation Theory 85
Exercises 89
xii Contents
2.2 The Schwartz Class and the Fourier Transform 94
2.2.1 The Class of Schwartz Functions 95
2.2.2 The Fourier Transform of a Schwartz Function 98
2.2.3 The Inverse Fourier Transform and Fourier Inversion 102
2.2.4 The Fourier Transform on Ll +L2 103
Exercises 106
2.3 The Class of Tempered Distributions 109
2.3.1 Spaces of Test Functions 109
2.3.2 Spaces of Functionals on Test Functions 110
2.3.3 The Space of Tempered Distributions 112
2.3.4 The Space of Tempered Distributions Modulo Polynomials .. 121
Exercises 122
2.4 More About Distributions and the Fourier Transform 124
2.4.1 Distributions Supported at a Point 124
2.4.2 The Laplacian 125
2.4.3 Homogeneous Distributions 127
Exercises 133
2.5 Convolution Operators on IP Spaces and Multipliers 135
2.5.1 Operators That Commute with Translations 135
2.5.2 The Transpose and the Adjoint of a Linear Operator 138
2.5.3 The Spaces ^P^(R ) 139
2.5.4 Characterizations of Jtx-X(R ) and ^2 2(R ) 141
2.5.5 The Space of Fourier Multipliers J?P(R ) 143
Exercises 146
2.6 Oscillatory Integrals 148
2.6.1 Phases with No Critical Points 149
2.6.2 Sublevel Set Estimates and the Van der Corput Lemma 151
Exercises 156
3 Fourier Analysis on the Torus 161
3.1 Fourier Coefficients 161
3.1.1 The n-Torus T 162
3.1.2 Fourier Coefficients 163
3.1.3 The Dirichlet and Fejer Kernels 165
3.1.4 Reproduction of Functions from Their Fourier Coefficients ..168
3.1.5 The Poisson Summation Formula 171
Exercises 173
3.2 Decay of Fourier Coefficients 176
3.2.1 Decay of Fourier Coefficients of Arbitrary Integrable
Functions 176
3.2.2 Decay of Fourier Coefficients of Smooth Functions 179
3.2.3 Functions with Absolutely Summable Fourier Coefficients.. 183
Exercises 185
3.3 Pointwise Convergence of Fourier Series 186
3.3.1 Pointwise Convergence of the Fejer Means 186
Contents xiii
3.3.2 Almost Everywhere Convergence of the Fejer Means 188
3.3.3 Pointwise Divergence of the Dirichlet Means 191
3.3.4 Pointwise Convergence of the Dirichlet Means 192
Exercises 193
3.4 Divergence of Fourier and Bochner-Riesz Summability 195
3.4.1 Motivation for Bochner-Riesz Summability 195
3.4.2 Divergence of Fourier Series of Integrable Functions 198
3.4.3 Divergence of Bochner-Riesz Means of Integrable
Functions 203
Exercises 209
3.5 The Conjugate Function and Convergence in Norm 211
3.5.1 Equivalent Formulations of Convergence in Norm 211
3.5.2 The U Boundedness of the Conjugate Function 215
Exercises 218
3.6 Multipliers, Transference, and Almost Everywhere Convergence .... 220
3.6.1 Multipliers on the Torus 221
3.6.2 Transference of Multipliers 223
3.6.3 Applications of Transference 228
3.6.4 Transference of Maximal Multipliers 228
3.6.5 Transference and Almost Everywhere Convergence 232
Exercises 235
3.7 Lacunary Series 237
3.7.1 Definition and Basic Properties of Lacunary Series 238
3.7.2 Equivalence of IP Norms of Lacunary Series 240
Exercises 245
4 Singular Integrals of Convolution Type 249
4.1 The Hilbert Transform and the Riesz Transforms 249
4.1.1 Definition and Basic Properties of the Hilbert Transform ... 250
4.1.2 Connections with Analytic Functions 253
4.1.3 LP Boundedness of the Hilbert Transform 255
4.1.4 The Riesz Transforms 259
Exercises 263
4.2 Homogeneous Singular Integrals and the Method of Rotations 267
4.2.1 Homogeneous Singular and Maximal Singular Integrals .... 267
4.2.2 Lr Boundedness of Homogeneous Singular Integrals 269
4.2.3 The Method of Rotations 272
4.2.4 Singular Integrals with Even Kernels 274
4.2.5 Maximal Singular Integrals with Even Kernels 278
Exercises 284
4.3 The Calderon-Zygmund Decomposition and Singular Integrals.... 286
4.3.1 The Calderon-Zygmund Decomposition 286
4.3.2 General Singular Integrals 289
4.3.3 U Boundedness Implies Weak Type (1.1) Boundedness ... 290
4.3.4 Discussion on Maximal Singular Integrals 293
xiv Contents
4.3.5 Boundedness for Maximal Singular Integrals Implies
Weak Type (1,1) Boundedness 297
Exercises 302
4.4 Sufficient Conditions for LP Boundedness 305
4.4.1 Sufficient Conditions for LP Boundedness of Singular
Integrals 305
4.4.2 An Example 308
4.4.3 Necessity of the Cancellation Condition 309
4.4.4 Sufficient Conditions for LP Boundedness of Maximal
Singular Integrals 310
Exercises 314
4.5 Vector-Valued Inequalities 315
4.5.1 £2-Valued Extensions of Linear Operators 316
4.5.2 Applications and lr-Valued Extensions of Linear Operators ..319
4.5.3 General Banach-Valued Extensions 321
Exercises 327
4.6 Vector-Valued Singular Integrals 329
4.6.1 Banach-Valued Singular Integral Operators 329
4.6.2 Applications 332
4.6.3 Vector-Valued Estimates for Maximal Functions 334
Exercises 337
5 Littlewood-Paley Theory and Multipliers 341
5.1 Littlewood-Paley Theory 341
5.1.1 The Littlewood-Paley Theorem 342
5.1.2 Vector-Valued Analogues 347
5.1.3 LP Estimates for Square Functions Associated with Dyadic
Sums 348
5.1.4 Lack of Orthogonality on LP 353
Exercises 355
5.2 Two Multiplier Theorems 359
5.2.1 The Marcinkiewicz Multiplier Theorem on R 360
5.2.2 The Marcinkiewicz Multiplier Theorem on R 363
5.2.3 The Hormander-Mihlin Multiplier Theorem on R 366
Exercises 371
5.3 Applications of Littlewood-Paley Theory 373
5.3.1 Estimates for Maximal Operators 373
5.3.2 Estimates for Singular Integrals with Rough Kernels 375
5.3.3 An Almost Orthogonality Principle on LP 379
Exercises 381
5.4 The Haar System, Conditional Expectation, and Martingales 383
5.4.1 Conditional Expectation and Dyadic Martingale Differences.. 384
5.4.2 Relation Between Dyadic Martingale Differences and
Haar Functions 385
5.4.3 The Dyadic Martingale Square Function 388
Contents xv
5.4.4 Almost Orthogonality Between the Littlewood-Paley
Operators and the Dyadic Martingale Difference Operators... 391
Exercises 394
5.5 The Spherical Maximal Function 395
5.5.1 Introduction of the Spherical Maximal Function 395
5.5.2 The First Key Lemma 397
5.5.3 The Second Key Lemma 399
5.5.4 Completion of the Proof 400
Exercises 400
5.6 Wavelets 402
5.6.1 Some Preliminary Facts 403
5.6.2 Construction of a Nonsmooth Wavelet 404
5.6.3 Construction of a Smooth Wavelet 406
5.6.4 A Sampling Theorem 410
Exercises 411
A Gamma and Beta Functions 417
A. 1 A Useful Formula 417
A.2 Definitions of JT(z) and B(z, w) 417
A.3 Volume of the Unit Ball and Surface of the Unit Sphere 418
A.4 Computation of Integrals Using Gamma Functions 419
A.5 Meromorphic Extensions of B(z, w) and F(z) 420
A.6 Asymptotics ofT(jc) as x — « 420
A.7 Euler s Limit Formula for the Gamma Function 421
A.8 Reflection and Duplication Formulas for the Gamma Function 424
B Bessel Functions 425
B.I Definition 425
B.2 Some Basic Properties 425
B.3 An Interesting Identity 427
B.4 The Fourier Transform of Surface Measure on S 1 428
B.5 The Fourier Transform of a Radial Function on R 428
B.6 Bessel Functions of Small Arguments 429
B.7 Bessel Functions of Large Arguments 430
B.8 Asymptotics of Bessel Functions 431
C Rademacher Functions 435
C. 1 Definition of the Rademacher Functions 435
C.2 Khintchine s Inequalities 435
C.3 Derivation of Khintchine s Inequalities 436
C.4 Khintchine s Inequalities for Weak Type Spaces 438
C.5 Extension to Several Variables 439
xvi Contents
D Spherical Coordinates 441
D. 1 Spherical Coordinate Formula 441
D.2 A Useful Change of Variables Formula 441
D.3 Computation of an Integral over the Sphere 442
D.4 The Computation of Another Integral over the Sphere 443
D.5 Integration over a General Surface 444
D.6 The Stereographic Projection 444
E Some Trigonometric Identities and Inequalities 447
F Summation by Parts 449
G Basic Functional Analysis 451
H The Minimax Lemma 453
I The Schur Lemma 457
1.1 The Classical Schur Lemma 457
1.2 Schur s Lemma for Positive Operators 457
1.3 An Example 460
J The Whitney Decomposition of Open Sets in R 463
K Smoothness and Vanishing Moments 465
K.I The Case of No Cancellation 465
K.2 The Case of Cancellation 466
K.3 The Case of Three Factors 467
Glossary 469
References 473
Index 485
|
| any_adam_object | 1 |
| author | Grafakos, Loukas 1962- |
| author_GND | (DE-588)136654444 |
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| callnumber-search | QA403.5 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 450 |
| classification_tum | MAT 420f |
| ctrlnum | (OCoLC)233933464 (DE-599)DNB988365251 |
| dewey-full | 515.2433 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.2433 |
| dewey-search | 515.2433 |
| dewey-sort | 3515.2433 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| era | Geschichte 1949-1978 gnd Geschichte 1949 gnd |
| era_facet | Geschichte 1949-1978 Geschichte 1949 |
| format | Book |
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| geographic | China (DE-588)4009937-4 gnd |
| geographic_facet | China |
| id | DE-604.BV035085939 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:19:49Z |
| institution | BVB |
| isbn | 9780387094311 9780387094328 9781441918550 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016754122 |
| oclc_num | 233933464 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-384 DE-706 DE-29T DE-92 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-824 DE-355 DE-BY-UBR DE-898 DE-BY-UBR |
| owner_facet | DE-703 DE-20 DE-384 DE-706 DE-29T DE-92 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-824 DE-355 DE-BY-UBR DE-898 DE-BY-UBR |
| physical | XVI, 489 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spellingShingle | Grafakos, Loukas 1962- Classical Fourier analysis Graduate texts in mathematics Fourier, Analyse de Fourier analysis Wirtschaftsentwicklung (DE-588)4066438-7 gnd Harmonische Analyse (DE-588)4023453-8 gnd Wirtschaftspolitik (DE-588)4066493-4 gnd Sozialer Wandel (DE-588)4077587-2 gnd |
| subject_GND | (DE-588)4066438-7 (DE-588)4023453-8 (DE-588)4066493-4 (DE-588)4077587-2 (DE-588)4009937-4 |
| title | Classical Fourier analysis |
| title_auth | Classical Fourier analysis |
| title_exact_search | Classical Fourier analysis |
| title_full | Classical Fourier analysis Loukas Grafakos |
| title_fullStr | Classical Fourier analysis Loukas Grafakos |
| title_full_unstemmed | Classical Fourier analysis Loukas Grafakos |
| title_old | Grafakos, Loukas Classical and modern Fourier analysis |
| title_short | Classical Fourier analysis |
| title_sort | classical fourier analysis |
| topic | Fourier, Analyse de Fourier analysis Wirtschaftsentwicklung (DE-588)4066438-7 gnd Harmonische Analyse (DE-588)4023453-8 gnd Wirtschaftspolitik (DE-588)4066493-4 gnd Sozialer Wandel (DE-588)4077587-2 gnd |
| topic_facet | Fourier, Analyse de Fourier analysis Wirtschaftsentwicklung Harmonische Analyse Wirtschaftspolitik Sozialer Wandel China |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754122&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT grafakosloukas classicalfourieranalysis |
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