Representation theorems in Hardy spaces:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge University Press
2009
|
| Ausgabe: | 1. publication |
| Schriftenreihe: | London Mathematical Society Student Texts
74 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017657757&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 372 Seitem graph. Darst. 23 cm |
| ISBN: | 9780521732017 9780521517683 |
Internformat
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| 100 | 1 | |a Mashreghi, Javad |d 1968- |e Verfasser |0 (DE-588)138145431 |4 aut | |
| 245 | 1 | 0 | |a Representation theorems in Hardy spaces |c Javad Mashreghi, Université Laval |
| 250 | |a 1. publication | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge University Press |c 2009 | |
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Datensatz im Suchindex
| _version_ | 1819347659989712896 |
|---|---|
| adam_text | Titel: Representation theorems in Hardy spaces
Autor: Mashreghi, Javad
Jahr: 2009
Contents
Preface xi
1 Fourier series 1
1.1 The Laplacian............................. 1
1.2 Some function spaces and sequence spaces............. 5
1.3 Fourier coefficients.......................... 8
1.4 Convolution on T........................... 13
1.5 Young s inequality.......................... 16
2 Abel—Poisson means 21
2.1 Abel-Poisson means of Fourier series................ 21
2.2 Approximate identities on T..................... 25
2.3 Uniform convergence and pointwise convergence.......... 32
2.4 Weak* convergence of measures................... 39
2.5 Convergence in norm......................... 43
2.6 Weak* convergence of bounded functions ............. 47
2.7 Parseval s identity.......................... 49
3 Harmonic functions in the unit disc 55
3.1 Series representation of harmonic functions............ 55
3.2 Hardy spaces on D.......................... 59
3.3 Poisson representation of h°° (D) functions............. 60
3.4 Poisson representation of hp(H) functions (1 p oo) ....... 65
3.5 Poisson representation of h1 (E ) functions ..... ......... 66
3.6 Radial limits of /ip(D) functions (1 p oo)........... 70
3.7 Series representation of the harmonic conjugate.......... 77
4 Logarithmic convexity 81
4.1 Subharmonic functions........................ 81
4.2 The maximum principle....................... 84
4.3 A characterization of subharmonic functions............ 88
4.4 Various means of subharmonic functions.............. 90
4.5 Radial subharmonic functions.................... 95
4.6 Hardy s convexity theorem...................... 97
vu
vin
CONTENTS
4.7 A complete characterization of hp(B) spaces............ 99
5 Analytic functions in the unit disc 103
5.1 Representation of Hp(B) functions (1 p oo)..........103
5.2 The Hubert transform on T.....................106
5.3 Radial limits of the conjugate function...............110
5.4 The Hilbert transform of C^T) functions ......:.;.... 113
5.5 Analytic measures on T.......................116
5.6 Representations of H1^) functions.................120
5.7 The uniqueness theorem and its applications ...........123
6 Norm inequalities for the conjugate function 131
6.1 Kolmogorov s theorems .......................131
6.2 Harmonic conjugate of h2(B) functions...............135
6.3 M. Riesz s theorem......................... 136
6.4 The Hilbert transform of bounded functions . . ..........142
6.5 The Hilbert transform of Dini continuous functions........144
6.6 Zygmund s LlogZ, theorem..................... 149
6.7 M. Riesz s LlogL theorem...................... 153
7 Blaschke products and their applications 155
7.1 Automorphisms of the open unit disc................155
7.2 Blaschke products for the open unit disc..............158
7.3 Jensen s formula............... .........162
7.4 Riesz s decomposition theorem...................166
7.5 Representation of HP(B) functions (0 p 1).......... 168
7.6 The canonical factorization in HP(B) (0 p oo) ........ 172
7.7 The Nevanlinna class.........................175
7.8 The Hardy and Fejér-Riesz inequalities................181
8 Interpolating linear operators 187
8.1 Operators on Lebesgue spaces....................187
8.2 Hadamard s three-line theorem...................189
8.3 The Riesz-Thorin interpolation theorem.............. 191
8.4 The Hausdorff-Young theorem...................197
8.5 An interpolation theorem for Hardy spaces............ 200
8.6 The Hardy-Littlewood inequality..................205
9 The Fourier transform 207
9.1 Lebesgue spaces on the real line.................. 207
9.2 The Fourier transform on L^E)...... . .........209
9.3 The multiplication formula on LX(R).............. ... 218
9.4 Convolution on R........ 219
9.5 Young s inequality....................... 221
CONTENTS ix
10 Poisson integrals 225
10.1 An application of the multiplication formula on L1 (R)...... 225
10.2 The conjugate Poisson kernel.................... 227
10.3 Approximate identities on E..................... 229
10.4 Uniform convergence and pointwise convergence.......... 232
10.5 Weak* convergence of measures . . . ................ 238
10.6 Convergence in norm......................... 241
10.7 Weak* convergence of bounded functions............. 243
11 Harmonic functions in the upper half plane 247
11.1 Hardy spaces on C+......................... 247
11.2 Poisson representation for semidiscs................ 248
11.3 Poisson representation of h(C+) functions............. 250
11.4 Poisson representation of hp(C+) functions (1 p oo)..... 252
11.5 A correspondence between C+ and E ................ 253
11.6 Poisson representation of positive harmonic functions....... 255
11.7 Vertical limits of hp(C+) functions (1 p oo).......... 258
12 The Plancherel transform 263
12.1 The inversion formula........................ 263
12.2 The Fourier-Plancherel transform.................. 266
12.3 The multiplication formula on LP(R) (1 p 2).........271
12.4 The Fourier transform on D (R) (l p 2)............ 273
12.5 An application of the multiplication formula on LP(R) (1 p 2) 274
12.6 A complete characterization of hp(C+) spaces........... 276
13 Analytic functions in the upper half plane 279
13.1 Representation of HP(C+) functions (1 p oo)......... 279
13.2 Analytic measures on R....................... 284
13.3 Representation of i/1^) functions................ 286
13.4 Spectral analysis of HP(R) (1 p 2)............... 287
13.5 A contraction from HP(C+) into J¥P(D).............. 289
13.6 Blaschke products for the upper half plane............. 293
13.7 The canonical factorization in HP(C+) (0 p oo) ....... 294
13.8 A correspondence between HP(C+) and HP(B).......... 298
14 The Hubert transform on R 301
14.1 Various definitions of the Hubert transform............ 301
14.2 The Hubert transform of C (R) functions............. 303
14.3 Almost everywhere existence of the Hubert transform...... 305
14.4 Kolmogorov s theorem........................ 308
14.5 M. Riesz s theorem.......................... 311
14.6 The Hubert transform of LipQ(t) functions............. 321
14.7 Maximal functions.......................... 329
14.8 The maximal Hubert transform................... 336
x CONTENTS
A Topics from real analysis 339
A.I A very concise treatment of measure theory............. 339
A.2 Riesz representation theorems..................... 344
A.3 Weak* convergence of measures................... 345
A.4 C(T) is dense in LP(T) (0 p oo)................. 346
A.5 The distribution function...................... 347
A.6 Minkowski s inequality........................ 348
A.7 Jensen s inequality......................... . 349
B A panoramic view of the representation theorems 351
B.I hp(B) ................................. . 352
B.I.I tfÇD)........................... ... 352
B.1.2 hp(B) (1 p oo)..................... . 354
B.1.3 h°°(B)............................. 355
B.2 HP(3).................................356
B.2.1 HP(B) (1 p oo) ......¦.... ...........356
B.2.2 H°°(B)............................358
B.3 hp(C+)................................ 359
B.3.1 fe1(C+) ............................359
B.3.2 hp(C+) (1 p 2)......................361
B.3.3 hP(C+) (2 p oo).....................362
B.3.4 h°°(C+)............................363
B.3.5 h+(C+)............................363
B.4 Hp(C+)................................364
B.4.1 Hp(C+) (1 p 2).....................364
B.4.2 HP(C+) (2 p oo) . . . ..................365
B.4.3 iï°°(C+) ...........................366
Bibliography 367
Index 369
|
| any_adam_object | 1 |
| author | Mashreghi, Javad 1968- |
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| dewey-raw | 515.94 |
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| discipline | Mathematik |
| edition | 1. publication |
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| id | DE-604.BV035602736 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:38:46Z |
| institution | BVB |
| isbn | 9780521732017 9780521517683 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017657757 |
| oclc_num | 244653541 |
| open_access_boolean | |
| owner | DE-824 DE-188 DE-11 DE-703 DE-384 |
| owner_facet | DE-824 DE-188 DE-11 DE-703 DE-384 |
| physical | XII, 372 Seitem graph. Darst. 23 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge University Press |
| record_format | marc |
| series | London Mathematical Society Student Texts |
| series2 | London Mathematical Society Student Texts |
| spellingShingle | Mashreghi, Javad 1968- Representation theorems in Hardy spaces London Mathematical Society Student Texts Analytic functions Hardy spaces Integralformel (DE-588)4161910-9 gnd Hardy-Raum (DE-588)4159109-4 gnd |
| subject_GND | (DE-588)4161910-9 (DE-588)4159109-4 |
| title | Representation theorems in Hardy spaces |
| title_auth | Representation theorems in Hardy spaces |
| title_exact_search | Representation theorems in Hardy spaces |
| title_full | Representation theorems in Hardy spaces Javad Mashreghi, Université Laval |
| title_fullStr | Representation theorems in Hardy spaces Javad Mashreghi, Université Laval |
| title_full_unstemmed | Representation theorems in Hardy spaces Javad Mashreghi, Université Laval |
| title_short | Representation theorems in Hardy spaces |
| title_sort | representation theorems in hardy spaces |
| topic | Analytic functions Hardy spaces Integralformel (DE-588)4161910-9 gnd Hardy-Raum (DE-588)4159109-4 gnd |
| topic_facet | Analytic functions Hardy spaces Integralformel Hardy-Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017657757&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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