Explorations in harmonic analysis: with applications to complex function theory and the Heisenberg group
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2009
|
| Schriftenreihe: | Applied and numerical harmonic analysis
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 360 S. Ill., graph. Darst. |
| ISBN: | 9780817646691 9780817646684 081764668X |
Internformat
MARC
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|---|---|---|---|
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| 020 | |a 9780817646691 |9 978-0-8176-4669-1 | ||
| 020 | |a 9780817646684 |c Gb. (Pr. in Vorb.) |9 978-0-8176-4668-4 | ||
| 020 | |a 081764668X |c Gb. (Pr. in Vorb.) |9 0-8176-4668-X | ||
| 024 | 3 | |a 9780817646684 | |
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| 035 | |a (OCoLC)233790518 | ||
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| 100 | 1 | |a Krantz, Steven G. |d 1951- |e Verfasser |0 (DE-588)130535907 |4 aut | |
| 245 | 1 | 0 | |a Explorations in harmonic analysis |b with applications to complex function theory and the Heisenberg group |c Steven G. Krantz ; with the assistance of Lina Lee |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2009 | |
| 300 | |a XIV, 360 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Applied and numerical harmonic analysis | |
| 650 | 4 | |a Harmonic analysis | |
| 650 | 0 | 7 | |a Harmonische Analyse |0 (DE-588)4023453-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Abstrakte harmonische Analysis |0 (DE-588)4821471-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Harmonische Analyse |0 (DE-588)4023453-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Abstrakte harmonische Analysis |0 (DE-588)4821471-1 |D s |
| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Lee, Lina |e Sonstige |4 oth | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017611324&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017611324 | ||
Datensatz im Suchindex
| _version_ | 1804139202225373184 |
|---|---|
| adam_text | Contents
Preface
Xl
1
Ontology and History of Real Analysis
.......................... 1
1.1
Deep Background in Real Analytic Functions
................... 1
1.2
The Idea of Fourier Expansions
............................... 3
1.3
Differences between the Real Analytic Theory and the Fourier
Theory
................................................... 5
1.4
Modem Developments
...................................... 6
1.5
Wavelets and Beyond
....................................... 6
1.6
History and Genesis of Fourier Series
.......................... 7
1.6.1
Derivation of the Heat Equation
........................ 10
2
The Central Idea: The Hilbert Transform
........................
IS
2.1
The Notion of the Hilbert Transform
.......................... 16
2.2
The Guts of the Hilbert Transform
............................ 17
2.3
The Laplace Equation and Singular Integrals
.................... 19
2.4
Boundedness of the Hilbert Transform
......................... 21
2.5
LP Boundedness of the Hilbert Transform
...................... 28
2.6
The Modified Hilbert Transform
.............................. 29
3
Essentíais
of the Fourier Transform
............................. 35
3.1
Quadratic Integrals and Plancherel
............................ 35
3.2
Sobolev Space Basics
....................................... 37
3.3
Key Concepts of Fractional Integrals
.......................... 41
3.4
The Sense of Singular Integrals
............................... 43
3.5
Ideas Leading to Pseudodifferential Operators
................... 45
4
Fractional and Singular Integrals
............................... 49
4.1
Fractional and Singular Integrals Together
...................... 49
4.2
Fractional Integrals
......................................... 51
4.3
Lead-in to Singular Integral Theory
........................... 53
viii Contents
5
Several Complex Variables
.................................... 61
5.1
What Is a Holomorphic Function?
............................. 62
5.2
Plurisubharmonic Functions
.................................. 64
5.3
Basic Concepts of Convexity
................................. 68
5.3.1
The Analytic Definition of Convexity
................... 70
5.3.2
Convexity with Respect to a Family of Functions
.......... 74
5.3.3
A Complex Analogue of Convexity
..................... 75
5.3.4
Further Remarks about Pseudoconvexity
................. 80
6
Pseudoconvexity and Domains of Holomorphy
.................... 83
6.1
Comparing Convexity and Pseudoconvexity
.................... 83
6.
1
.1
Holomorphic Support Functions
........................ 87
6.1.2
Peaking Functions
................................... 91
6.2
Pseudoconvexity and Analytic Disks
.......................... 92
6.3
Domains of Holomorphy
.................................... 102
6.3.1
Consequences of Theorems
6.2.5
and
6.3.6............... 106
6.3.2
Consequences of the
Levi
Problem
..................... 107
7
Canonical Complex Integral Operators
..........................
Ill
7.
1 Elementary Concepts of the Bergman Kernel
................... 112
7.1.1
Smoothness to the Boundary of
K q
..................... 121
7.1.2
Calculating the Bergman Kernel
........................
1
22
7.2
The
Szegő
Kernel
.......................................... 127
8
Hardy Spaces Old and New
....................................
1
33
8.1
Hardy Spaces on the Unit Disk
............................... 134
8.2
Key Properties of the
Poisson
Kernel
.......................... 142
8.3
The Centrality of Subharmonicity
............................. 145
8.4
More about Pointwise Convergence
........................... 152
8.5
A Preliminary Result in Complex Domains
..................... 155
8.6
First Concepts of Admissible Convergence
..................... 156
8.7
Real-Variable Methods
...................................... 167
8.8
Real-Variable Hardy Spaces
.................................. 168
8.9
Maximal Functions and Hardy Spaces
......................... 171
8.10
The Atomic Theory of Hardy Spaces
.......................... 172
8.11
The Role of BMO
.......................................... 174
9
Introduction to the
Heisenberg
Group
........................... 179
9.1
The Classical Upper Half-Plane
............................... 179
9.2
Background in Quantum Mechanics
........................... 181
9.3
The Role of the
Heisenberg
Group
............................ 182
9.4
The
Heisenberg
Group and Its Action on
Ы
..................... 184
9.5
The Geometry of
õU
........................................ 1
g7
9.6
The Lie Group Structure of H
............................... 188
Contents ix
9.6.1
Distinguished
1 -Parameter
Subgroups of the
Heisenberg
Group
.............................................. 188
9.6.2
Commutators of Vector Fields
......................... 189
9.6.3
Commutators in the
Heisenberg
Group
.................. 191
9.6.4
Additional Information about the
Heisenberg
Group Action
. 191
9.7
A Fresh Look at Classical Analysis
............................ 192
9.7.1
Spaces of Homogeneous Type
......................... 192
9.7.2
The Folland-Stein Theorem
........................... 194
9.7.3
Classical
Calderón-Zygmund
Theory
................... 198
9.8
Analysis on H
............................................ 209
9.8.1
The Norm on H
.................................... 210
9.8.2
Polar Coordinates
.................................... 211
9.8.3
Some Remarks about Hausdorff Measure
................ 212
9.8.4
Integration in
И
.................................... 213
9.8.5
Distance in H
...................................... 214
9.8.6
H Is a Space of Homogeneous Type
.................... 214
- 9.8.7
Homogeneous Functions
.............................. 215
9.9
Boundedness of Singular Integrals on I?
....................... 217
9.9.1
Cotlar-Knapp-Stein Lemma
........................... 218
9.9.2
The Folland-Stein Theorem
........................... 220
9.10
Boundedness of Singular Integrals onL
....................... 228
9.11
Remarks on
W
and BMO
................................... 228
10
Analysis on the
Heisenberg
Group
..............................231
10.1
The
Szegő
Kernel on the
Heisenberg
Group
.................... 232
10.2
The Poisson-Szego Kernel on the
Heisenberg
Group
............. 232
10.3
Kernels on the
Siegel
Space
.................................. 233
10.3.1
Sets of Determinacy
.................................. 233
10.3.2
The
Szegő
Kernel on the
Siegel
Upper Half-Space
W
...... 234
11
A Coda on Domains of Finite Type
..............................249
11.1
Prefatory Remarks
___._..................................... 249
11.1.1
The Role of the
ô
Problem
............................ 250
11.2
Return to Finite Type
....................................... 253
11.3
Finite Type in Dimension Two
................................ 255
1
1.4
Finite Type in Higher Dimensions
............................. 259
Appendix
1:
Rudiments of Fourier Series
............................ 265
A
1.1
Fourier Series: Fundamental Ideas
............................ 265
A1.2 Basics
....................................................
268
A1
.3
Summability Methods
.......................................
27
1
A
1.4
Ideas from Elementary Functional Analysis
..................... 275
A1.5 Summability Kernels
.......................................
277
A
1.6
Pointwise Convergence
...................................... 282
A
1.7
Square Integrals
............................................
2**3
χ
Contents
Appendix 2:
The Fourier Transform
................................297
A2.
1
Fundamental Properties of the Fourier Transform
................ 297
A2.2
Invariance
and Symmetry Properties
........................... 299
A2.3 Convolution and Fourier Inversion
............................ 303
A2.3.
1
The Inverse Fourier Transform
......................... 304
Appendix
3:
Pseudodifferential Operators
...........................
31S
A3.1
Introduction to the Operators
................................. 315
A3.2 A Formal Treatment
........................................ 320
A3.3 The Calculus of Operators
................................... 325
References
......................................................337
Index
...........................................................349
|
| any_adam_object | 1 |
| author | Krantz, Steven G. 1951- |
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.2433 |
| dewey-sort | 3515 42433 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035555491 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:40:20Z |
| institution | BVB |
| isbn | 9780817646691 9780817646684 081764668X |
| language | English |
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| physical | XIV, 360 S. Ill., graph. Darst. |
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| publisher | Birkhäuser |
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| series2 | Applied and numerical harmonic analysis |
| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group Steven G. Krantz ; with the assistance of Lina Lee Boston [u.a.] Birkhäuser 2009 XIV, 360 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis Harmonic analysis Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Abstrakte harmonische Analysis (DE-588)4821471-1 gnd rswk-swf Harmonische Analyse (DE-588)4023453-8 s DE-604 Abstrakte harmonische Analysis (DE-588)4821471-1 s Lee, Lina Sonstige oth Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017611324&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group Harmonic analysis Harmonische Analyse (DE-588)4023453-8 gnd Abstrakte harmonische Analysis (DE-588)4821471-1 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4821471-1 |
| title | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group |
| title_auth | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group |
| title_exact_search | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group |
| title_full | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group Steven G. Krantz ; with the assistance of Lina Lee |
| title_fullStr | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group Steven G. Krantz ; with the assistance of Lina Lee |
| title_full_unstemmed | Explorations in harmonic analysis with applications to complex function theory and the Heisenberg group Steven G. Krantz ; with the assistance of Lina Lee |
| title_short | Explorations in harmonic analysis |
| title_sort | explorations in harmonic analysis with applications to complex function theory and the heisenberg group |
| title_sub | with applications to complex function theory and the Heisenberg group |
| topic | Harmonic analysis Harmonische Analyse (DE-588)4023453-8 gnd Abstrakte harmonische Analysis (DE-588)4821471-1 gnd |
| topic_facet | Harmonic analysis Harmonische Analyse Abstrakte harmonische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017611324&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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