The analysis of linear partial differential operators: 1 Distribution theory and Fourier analysis
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
2003
|
| Edition: | Reprint of the 2nd edition 1990 |
| Series: | Classics in mathematics
Die Grundlehren der mathematischen Wissenschaften |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [419] - 436. - Originally published as Vol. 256 in the series: Grundlehren der mathematischen Wissenschaften in 1983 and 1990, and thereafter as a Springer study edition 1990 |
| Physical Description: | XI, 440 Seiten |
| ISBN: | 3540006621 9783540006626 |
Staff View
MARC
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| 245 | 1 | 0 | |a The analysis of linear partial differential operators |n 1 |p Distribution theory and Fourier analysis |c Lars Hörmander |
| 250 | |a Reprint of the 2nd edition 1990 | ||
| 264 | 1 | |a Berlin |b Springer |c 2003 | |
| 300 | |a XI, 440 Seiten | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Classics in mathematics | |
| 490 | 0 | |a Die Grundlehren der mathematischen Wissenschaften | |
| 500 | |a Literaturverz. S. [419] - 436. - Originally published as Vol. 256 in the series: Grundlehren der mathematischen Wissenschaften in 1983 and 1990, and thereafter as a Springer study edition 1990 | ||
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Record in the Search Index
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| adam_text | Contents
Introduction
....................... 1
Chapter I. Test Functions
................. 5
Summary
....................... 5
1.1.
A review of Differential Calculus
........... 5
1.2.
Existence of Test Functions
............. 14
1.3.
Convolution
.................... 16
1.4.
Cutoff Functions and Partitions of Unity
....... 25
Notes
......................... 31
Chapter II. Definition and Basic Properties of Distributions
. . 33
Summary
....................... 33
2.1.
Basic Definitions
.................. 33
2.2.
Localization
.................... 41
2.3.
Distributions with Compact Support
......... 44
Notes
......................... 52
Chapter III. Differentiation and Multiplication by Functions
. . 54
Summary
....................... 54
3.1.
Definition and Examples
.............. 54
3.2.
Homogeneous Distributions
............. 68
3.3.
Some Fundamental Solutions
............ 79
3.4.
Evaluation of Some Integrals
............. 84
Notes
......................... 86
Chapter IV. Convolution
................. 87
Summary
....................... 87
4.1.
Convolution with a Smooth Function
......... 88
4.2.
Convolution of Distributions
............. 100
4.3.
The Theorem of Supports
.............. 105
4.4.
The Role of Fundamental Solutions
.......... 109
X
Contents
4.5.
Basic ¿/Estimates for Convolutions
..........116
Notes
.........................
124
Chapter V. Distributions in Product Spaces
.........126
Summary
....................... 126
5.1.
Tensor Products
.................. 126
5.2.
The Kernel Theorem
................ 128
Notes
......................... 132
Chapter VI. Composition with Smooth Maps
.........133
Summary
....................... 133
6.1.
Definitions
..................... 133
6.2.
Some Fundamental Solutions
............ 137
6.3.
Distributions on a Manifold
............. 142
6.4.
The Tangent and Cotangent Bundles
......... 146
Notes
......................... 156
Chapter
VII.
The Fourier Transformation
..........158
Summary
....................... 158
7.1.
The Fourier Transformation in
У
and in
У
...... 159
7.2.
Poisson s Summation Formula and Periodic Distributions
177
7.3.
The Fourier-Laplace Transformation in S
....... 181
7.4.
More General Fourier-Laplace Transforms
....... 191
7.5.
The
Malgrange
Preparation Theorem
......... 195
7.6.
Fourier Transforms of Gaussian Functions
....... 205
7.7.
The Method of Stationary Phase
........... 215
7.8.
Oscillatory Integrals
................ 236
7.9.
H(s), I? and Holder Estimates
............ 240
Notes
......................... 248
Chapter
VIII.
Spectral Analysis of Singularities
........251
Summary
.......................251
8.1.
The Wave Front Set
................252
8.2.
A Review of Operations with Distributions
......261
8.3.
The Wave Front Set of Solutions of Partial Differential
Equations
......................271
8.4.
The Wave Front Set with Respect to CL
........280
8.5.
Rules of Computation for WFL
............296
8.6.
WFL for Solutions of Partial Differential Equations
... 305
8.7.
Microhyperbolicity
.................317
Notes
.........................322
Contents
XI
Chapter IX. Hyperfunctions
................ 325
Summary
....................... 325
9.1.
Analytic Functionals
................ 326
9.2.
General Hyperfunctions
............... 335
9.3.
The Analytic Wave Front Set of a Hyperfunction
.... 338
9.4.
The Analytic Cauchy Problem
............ 346
9.5.
Hyperfunction Solutions of Partial Differential Equations
353
9.6.
The Analytic Wave Front Set and the Support
..... 358
Notes
......................... 368
Exercises
........................ 371
Answers and Hints to All the Exercises
............ 394
Bibliography
....................... 419
Index
.......................... 437
Index of Notation
..................... 439
|
| adam_txt |
Contents
Introduction
. 1
Chapter I. Test Functions
. 5
Summary
. 5
1.1.
A review of Differential Calculus
. 5
1.2.
Existence of Test Functions
. 14
1.3.
Convolution
. 16
1.4.
Cutoff Functions and Partitions of Unity
. 25
Notes
. 31
Chapter II. Definition and Basic Properties of Distributions
. . 33
Summary
. 33
2.1.
Basic Definitions
. 33
2.2.
Localization
. 41
2.3.
Distributions with Compact Support
. 44
Notes
. 52
Chapter III. Differentiation and Multiplication by Functions
. . 54
Summary
. 54
3.1.
Definition and Examples
. 54
3.2.
Homogeneous Distributions
. 68
3.3.
Some Fundamental Solutions
. 79
3.4.
Evaluation of Some Integrals
. 84
Notes
. 86
Chapter IV. Convolution
. 87
Summary
. 87
4.1.
Convolution with a Smooth Function
. 88
4.2.
Convolution of Distributions
. 100
4.3.
The Theorem of Supports
. 105
4.4.
The Role of Fundamental Solutions
. 109
X
Contents
4.5.
Basic ¿/Estimates for Convolutions
.116
Notes
.
124
Chapter V. Distributions in Product Spaces
.126
Summary
. 126
5.1.
Tensor Products
. 126
5.2.
The Kernel Theorem
. 128
Notes
. 132
Chapter VI. Composition with Smooth Maps
.133
Summary
. 133
6.1.
Definitions
. 133
6.2.
Some Fundamental Solutions
. 137
6.3.
Distributions on a Manifold
. 142
6.4.
The Tangent and Cotangent Bundles
. 146
Notes
. 156
Chapter
VII.
The Fourier Transformation
.158
Summary
. 158
7.1.
The Fourier Transformation in
У
and in
У
. 159
7.2.
Poisson's Summation Formula and Periodic Distributions
177
7.3.
The Fourier-Laplace Transformation in S"
. 181
7.4.
More General Fourier-Laplace Transforms
. 191
7.5.
The
Malgrange
Preparation Theorem
. 195
7.6.
Fourier Transforms of Gaussian Functions
. 205
7.7.
The Method of Stationary Phase
. 215
7.8.
Oscillatory Integrals
. 236
7.9.
H(s), I? and Holder Estimates
. 240
Notes
. 248
Chapter
VIII.
Spectral Analysis of Singularities
.251
Summary
.251
8.1.
The Wave Front Set
.252
8.2.
A Review of Operations with Distributions
.261
8.3.
The Wave Front Set of Solutions of Partial Differential
Equations
.271
8.4.
The Wave Front Set with Respect to CL
.280
8.5.
Rules of Computation for WFL
.296
8.6.
WFL for Solutions of Partial Differential Equations
. 305
8.7.
Microhyperbolicity
.317
Notes
.322
Contents
XI
Chapter IX. Hyperfunctions
. 325
Summary
. 325
9.1.
Analytic Functionals
. 326
9.2.
General Hyperfunctions
. 335
9.3.
The Analytic Wave Front Set of a Hyperfunction
. 338
9.4.
The Analytic Cauchy Problem
. 346
9.5.
Hyperfunction Solutions of Partial Differential Equations
353
9.6.
The Analytic Wave Front Set and the Support
. 358
Notes
. 368
Exercises
. 371
Answers and Hints to All the Exercises
. 394
Bibliography
. 419
Index
. 437
Index of Notation
. 439 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:37:31Z |
| indexdate | 2024-07-09T20:33:41Z |
| institution | BVB |
| isbn | 3540006621 9783540006626 |
| language | English |
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| series2 | Classics in mathematics Die Grundlehren der mathematischen Wissenschaften |
| spelling | Hörmander, Lars 1931-2012 Verfasser (DE-588)105823449 aut The analysis of linear partial differential operators 1 Distribution theory and Fourier analysis Lars Hörmander Reprint of the 2nd edition 1990 Berlin Springer 2003 XI, 440 Seiten txt rdacontent n rdamedia nc rdacarrier Classics in mathematics Die Grundlehren der mathematischen Wissenschaften Literaturverz. S. [419] - 436. - Originally published as Vol. 256 in the series: Grundlehren der mathematischen Wissenschaften in 1983 and 1990, and thereafter as a Springer study edition 1990 Linearer partieller Differentialoperator (DE-588)4167722-5 gnd rswk-swf Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd rswk-swf Linearer partieller Differentialoperator (DE-588)4167722-5 s DE-604 Lineare partielle Differentialgleichung (DE-588)4167708-0 s (DE-604)BV000157227 1 Erscheint auch als Online-Ausgabe 978-3-642-61497-2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014564935&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hörmander, Lars 1931-2012 The analysis of linear partial differential operators Linearer partieller Differentialoperator (DE-588)4167722-5 gnd Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd |
| subject_GND | (DE-588)4167722-5 (DE-588)4167708-0 |
| title | The analysis of linear partial differential operators |
| title_auth | The analysis of linear partial differential operators |
| title_exact_search | The analysis of linear partial differential operators |
| title_exact_search_txtP | The analysis of linear partial differential operators |
| title_full | The analysis of linear partial differential operators 1 Distribution theory and Fourier analysis Lars Hörmander |
| title_fullStr | The analysis of linear partial differential operators 1 Distribution theory and Fourier analysis Lars Hörmander |
| title_full_unstemmed | The analysis of linear partial differential operators 1 Distribution theory and Fourier analysis Lars Hörmander |
| title_short | The analysis of linear partial differential operators |
| title_sort | the analysis of linear partial differential operators distribution theory and fourier analysis |
| topic | Linearer partieller Differentialoperator (DE-588)4167722-5 gnd Lineare partielle Differentialgleichung (DE-588)4167708-0 gnd |
| topic_facet | Linearer partieller Differentialoperator Lineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014564935&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000157227 |
| work_keys_str_mv | AT hormanderlars theanalysisoflinearpartialdifferentialoperators1 |