Classical Fourier analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2014
|
| Edition: | Third edition |
| Series: | Graduate texts in mathematics
249 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | xvii, 638 Seiten Diagramme 235 mm x 155 mm |
| ISBN: | 9781493911936 9781493911943 |
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Record in the Search Index
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|---|---|
| adam_text | Loukas Grafakos
Classical Fourier Analysis
Third Edition
€l Springer
Contents
1 Z/Spaces and Interpolation 1
1 1 If and Weak LP 1
111 The Distribution Function 3
112 Convergence in Measure 6
113A First Glimpse at Interpolation 9
Exercises 11
1 2 Convolution and Approximate Identities 17
121 Examples of Topological Groups 18
122 Convolution 19
123 Basic Convolution Inequalities 21
124 Approximate Identities 25
Exercises 30
1 3 Interpolation 33
131 Real Method: The Marcinkiewicz Interpolation Theorem 33
132 Complex Method: The Riesz-Thorin Interpolation
Theorem : 36
133 Interpolation of Analytic Families of Operators 40
Exercises 45
1 4 Lorentz Spaces 48
141 Decreasing Rearrangements 48
142 Lorentz Spaces 52
143 Duals of Lorentz Spaces 56
144 The Off-Diagonal Marcinkiewicz Interpolation Theorem 60
Exercises 74 2
2 Maximal Functions, Fourier Transform, and Distributions 85
2 1 Maximal Functions 86
211 The Hardy-Littlewood Maximal Operator 86
212 Control of Other Maximal Operators 90
XI
Contents
xii
213 Applications to Differentiation Theory 93
Exercises 98
2 2 The Schwartz Class and the Fourier Transform 104
221 The Class of Schwartz Functions 105
222 The Fourier Transform of a Schwartz Function 108
223 The Inverse Fourier Transform and Fourier Inversion Ill
224 The Fourier Transform on L1 +L2 113
Exercises 116
2 3 The Class of Tempered Distributions 119
231 Spaces of Test Functions 119
232 Spaces of Functionals on Test Functions 120
233 The Space of Tempered Distributions 123
Exercises 131
2 4 More About Distributions and the Fourier Transform 133
241 Distributions Supported at a Point 134
242 The Laplacian 135
243 Homogeneous Distributions 136
Exercises 143
2 5 Convolution Operators on IP Spaces and Multipliers 146
251 Operators That Commute with Translations 146
252 The Transpose and the Adjoint of a Linear Operator 150
253 The Spaces R) 151
254 Characterizations of ^#1,1 (Rn) and ^2,2(Rn) 153
255 The Space of Fourier Multipliers ^p(Rn) 155
Exercises 159
2 6 Oscillatory Integrals 161
261 Phases with No Critical Points 161
262 Sublevel Set Estimates and the Van der Corput Lemma 164
Exercises 169 3
3 Fourier Series 173
3 1 Fourier Coefficients 173
311 The n-Torus Tn 174
312 Fourier Coefficients 175
313 The Dirichlet and Fejer Kernels 178
Exercises 182
3 2 Reproduction of Functions from Their Fourier Coefficients 183
321 Partial sums and Fourier inversion 183
322 Fourier series of square summable functions 185
323 The Poisson Summation Formula 187
Exercises 191
3 3 Decay of Fourier Coefficients 192
331 Decay of Fourier Coefficients of Arbitrary Integrable
Functions 193
332 Decay of Fourier Coefficients of Smooth Functions 195
Contents
xiii
333 Functions with Absolutely Summable Fourier
Coefficients 200
Exercises 202
3 4 Pointwise Convergence of Fourier Series 204
341 Pointwise Convergence of the Fejer Means 204
342 Almost Everywhere Convergence of the Fejer Means 207
343 Pointwise Divergence of the Dirichlet Means 210
344 Pointwise Convergence of the Dirichlet Means 212
Exercises 214
35A Tauberian theorem and Functions of Bounded Variation 216
351A Tauberian theorem 216
352 The sine integral function 218
353 Further properties of functions of bounded variation 219
354 Gibbs phenomenon 221
Exercises 225
3 6 Lacunary Series and Sidon Sets 226
361 Definition and Basic Properties of Lacunary Series 227
362 Equivalence of LP Norms of Lacunary Series 229
363 Sidon sets 235
Exercises 237 4
4 Topics on Fourier Series 241
4 1 Convergence in Norm, Conjugate Function,
and Bochner-Riesz Means 241
411 Equivalent Formulations of Convergence in Norm 242
412 The LP Boundedness of the Conjugate Function 246
413 Bochner-Riesz Summability 250
Exercises 253
42AE Divergence of Fourier Series and Bochner-Riesz means 255
421 Divergence of Fourier Series of Integrable Functions 255
422 Divergence of Bochner-Riesz Means of Integrable
Functions 261
Exercises 270
4 3 Multipliers, Transference, and Almost Everywhere Convergence 271
431 Multipliers on the Torus 271
432 Transference of Multipliers 275
433 Applications of Transference 280
434 Transference of Maximal Multipliers 281
435 Applications to Almost Everywhere Convergence 285
436 Almost Everywhere Convergence of Square Dirichlet
Means 287
Exercises 289
XIV
Contents
4 4 Applications to Geometry and Partial Differential Equations 292
441 The Isoperimetric Inequality 292
442 The Heat Equation with Periodic Boundary Condition 294
Exercises 298
4 5 Applications to Number theory and Ergodic theory 299
451 Evaluation of the Riemann Zeta Function at even Natural
numbers 299
452 Equidistributed sequences 302
453 The Number of Lattice Points inside a Ball 305
Exercises 308
5 Singular Integrals of Convolution Type 313
5 1 The Hilbert Transform and the Riesz Transforms 313
511 Definition and Basic Properties of the Hilbert Transform 314
512 Connections with Analytic Functions 317
513 LP Boundedness of the Hilbert Transform 319
514 The Riesz Transforms 324
Exercises 329
5 2 Homogeneous Singular Integrals and the Method of Rotations 333
521 Homogeneous Singular and Maximal Singular Integrals 333
522 L2 Boundedness of Homogeneous Singular Integrals 336
523 The Method of Rotations 339
524 Singular Integrals with Even Kernels 341
525 Maximal Singular Integrals with Even Kernels 347
Exercises 353
5 3 The Calderon-Zygmund Decomposition and Singular Integrals____355
531 The Calderon-Zygmund Decomposition 355
532 General Singular Integrals 358
533U Boundedness Implies Weak Type (1,1) Boundedness 359
534 Discussion on Maximal Singular Integrals 362
535 Boundedness for Maximal Singular Integrals Implies
Weak Type (1,1) Boundedness 366
Exercises 371
5 4 Sufficient Conditions for LP Boundedness 374
541 Sufficient Conditions for LP Boundedness of Singular
Integrals 375
542 An Example 378
543 Necessity of the Cancellation Condition 379
544 Sufficient Conditions for LP Boundedness of Maximal
Singular Integrals 380
Exercises 384
5 5 Vector-Valued Inequalities 385
551 ^-Valued Extensions of Linear Operators 386
552 Applications and ^-Valued Extensions of Linear
Operators 390
Contents
xv
553 General Banach-Valued Extensions 391
Exercises 398
5 6 Vector-Valued Singular Integrals 401
561 Banach-Valued Singular Integral Operators 402
562 Applications 408
563 Vector-Valued Estimates for Maximal Functions 411
Exercises 414
6 Littlewood-Paley Theory and Multipliers 419
6 1 Littlewood-Paley Theory 419
611 The Littlewood-Paley Theorem 420
612 Vector-Valued Analogues 426
613 LP Estimates for Square Functions Associated
with Dyadic Sums 426
614 Lack of Orthogonality on LP 431
Exercises 434
6 2 Two Multiplier Theorems 437
621 The Marcinkiewicz Multiplier Theorem on R 439
622 The Marcinkiewicz Multiplier Theorem on R 441
623 The Mihlin-Hörmander Multiplier Theorem onR 445
Exercises 450
6 3 Applications of Littlewood-Paley Theory 453
631 Estimates for Maximal Operators 453
632 Estimates for Singular Integrals with Rough Kernels 455
633 An Almost Orthogonality Principle on LP 459
Exercises 461
6 4 The Haar System, Conditional Expectation, and Martingales 463
641 Conditional Expectation and Dyadic Martingale
Differences 464
642 Relation Between Dyadic Martingale Differences
and Haar Functions 465
643 The Dyadic Martingale Square Function 469
644 Almost Orthogonality Between the Littlewood-Paley
Operators and the Dyadic Martingale Difference
Operators 471
Exercises 474
6 5 The Spherical Maximal Function 475
651 Introduction of the Spherical Maximal Function 475
652 The First Key Lemma 478
653 The Second Key Lemma 479
654 Completion of the Proof 481
Exercises 481
6 6 Wavelets and Sampling 482
661 Some Preliminary Facts 483
662 Construction of a Nonsmooth Wavelet 485
XVI
Contents
663 Construction of a Smooth Wavelet 486
664 Sampling 490
Exercises 494
7 Weighted Inequalities 499
7 1 The Ap Condition 499
711 Motivation for the Ap Condition 500
712 Properties of Ap Weights 503
Exercises 511
7 2 Reverse Holder Inequality for Ap Weights and Consequences 514
721 The Reverse Holder Property of Ap Weights 514
722 Consequences of the Reverse Holder Property 518
Exercises 521
7 3 The Aoc Condition 525
731 The Class of A« Weights 525
732 Characterizations of A«, Weights 527
Exercises 530
7 4 Weighted Norm Inequalities for Singular Integrals 532
741 Singular Integrals of Non Convolution type 532
742A Good Lambda Estimate for Singular Integrals 533
743 Consequences of the Good Lambda Estimate 539
744 Necessity of the Ap Condition 543
Exercises 545
7 5 Further Properties of Ap Weights 546
751 Factorization of Weights 546
752 Extrapolation from Weighted Estimates on a Single LPo 548
753 Weighted Inequalities Versus Vector-Valued Inequalities-554
Exercises 558
A Gamma and Beta Functions 563
AlA Useful Formula 563
A 2 Definitions of F(z) and B(z, w) 563
A 3 Volume of the Unit Ball and Surface of the Unit Sphere 565
A 4 Computation of Integrals Using Gamma Functions 565
A 5 Meromorphic Extensions of B(z,w) and F(z) 566
A 6 Asymptotics of F (x) as x -* °o 567
A 7 Euler’s Limit Formula for the Gamma Function 568
A 8 Reflection and Duplication Formulas for the Gamma Function 570
B Bessel Functions 573
B l Definition 573
B 2 Some Basic Properties 573
B 3 An Interesting Identity 576
B 4 The Fourier Transform of Surface Measure on S”1 577
B 5 The Fourier Transform of a Radial Function on R” 577
Contents xvii
B 6 Bessel Functions of Small Arguments 578
B 7 Bessel Functions of Large Arguments 579
B 8 Asymptotics of Bessel Functions 580
B 9 Bessel Functions of general complex indices 582
C Rademacher Functions 585
C l Definition of the Rademacher Functions 585
C 2 Khintchine’s Inequalities 586
C 3 Derivation of Khintchine’s Inequalities 586
C 4 Khintchine’s Inequalities for Weak Type Spaces 589
C 5 Extension to Several Variables 589
D Spherical Coordinates 591
D l Spherical Coordinate Formula 591
D2A Useful Change of Variables Formula 592
D 3 Computation of an Integral over the Sphere 593
D 4 The Computation of Another Integral over the Sphere 593
D 5 Integration over a General Surface 594
D 6 The Stereographic Projection 594
E Some Trigonometric Identities and Inequalities 597
F Summation by Parts 599
G Basic Functional Analysis 601
H The Minimax Lemma 603
I Taylor’s and Mean Value Theorem in Several Variables 607
1 1 Mutlivariable Taylor’s Theorem 607
1 2 The Mean value Theorem 608
J The Whitney Decomposition of Open Sets in R 609
J l Decomposition of Open Sets 609
J 2 Partition of Unity adapted to Whitney cubes 611
Glossary 613
References 617
Index
|
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| indexdate | 2024-07-10T01:15:50Z |
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| isbn | 9781493911936 9781493911943 |
| language | English |
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| spelling | Grafakos, Loukas 1962- (DE-588)136654444 aut Classical Fourier analysis Loukas Grafakos Third edition New York, NY Springer 2014 xvii, 638 Seiten Diagramme 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 249 Hier auch später erschienene, unveränderte Nachdrucke Fourier, Analyse de Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 Graduate texts in mathematics 249 (DE-604)BV000000067 249 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027666722&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Grafakos, Loukas 1962- Classical Fourier analysis Graduate texts in mathematics Fourier, Analyse de Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd |
| subject_GND | (DE-588)4023453-8 |
| 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_short | Classical Fourier analysis |
| title_sort | classical fourier analysis |
| topic | Fourier, Analyse de Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd |
| topic_facet | Fourier, Analyse de Fourier analysis Harmonische Analyse |
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