Analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2010
|
| Ausgabe: | Second edition, reprinted with corrections |
| Schriftenreihe: | Graduate studies in mathematics
14 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xxii, 348 Seiten |
| ISBN: | 0821827839 9780821827833 |
Internformat
MARC
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| 100 | 1 | |a Lieb, Elliott H. |d 1932- |e Verfasser |0 (DE-588)11899655X |4 aut | |
| 245 | 1 | 0 | |a Analysis |c Elliott H. Lieb ; Michael Loss |
| 250 | |a Second edition, reprinted with corrections | ||
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2010 | |
| 300 | |a xxii, 348 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 14 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Analysis | |
| 650 | 0 | 7 | |a Analysis |0 (DE-588)4001865-9 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Loss, Michael |d 1954- |e Verfasser |0 (DE-588)172513804 |4 aut | |
| 830 | 0 | |a Graduate studies in mathematics |v 14 |w (DE-604)BV009739289 |9 14 | |
| 856 | 4 | |u http://www.gbv.de/dms/goettingen/325084793.pdf |3 Inhaltsverzeichnis | |
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Datensatz im Suchindex
| _version_ | 1817975493915312128 |
|---|---|
| adam_text |
Contents
Preface
to the First Edition
xvii
Preface to the Second Edition
xxi
CHAPTER
1.
Measure and Integration
1
1.1
Introduction
1
1.2
Basic notions of measure theory
4
1.3
Monotone class theorem
9
1.4
Uniqueness of measures
11
1.5
Definition of measurable functions and integrals
12
1.6
Monotone convergence
17
1.7
Fatou's lemma
18
1.8
Dominated convergence
19
1.9
Missing term in Fatou's lemma
21
1.10
Product measure
23
1.11
Commutativity and associativity of product measures
24
1.12
Fubini's theorem
25
1.13
Layer cake representation
26
1.14
Bathtub principle
28
1.15
Constructing a measure from an outer measure
29
1.16
Uniform convergence except on small sets
31
1.17
Simple functions and really simple functions
32
1.18
Approximation by really simple functions
34
1.19 Approximation
by C°° functions
36
Exercises
37
CHAPTER
2.
¿P-Spaces
41
2.1
Definition of Z^-spaces
41
2.2
Jensen's inequality
44
2.3
Holder's inequality
45
2.4
Minkowski's inequality
47
2.5
Hanner's inequality
49
2.6
Differentiability of norms
51
2.7
Completeness of Lp-spaces
52
2.8
Projection on convex sets
53
2.9
Continuous linear functionals and weak convergence
54
2.10
Linear functionals separate
56
2.11
Lower semicontinuity of norms
57
2.12
Uniform boundedness principle
58
2.13
Strongly convergent convex combinations
60
2.14
The dual of
Ц>{П)
61
2.15
Convolution
64
2.16
Approximation by C°°-functions
64
2.17
Separability of IJP(Rn)
67
2.18
Bounded sequences have weak limits
68
2.19
Approximation by C^-functions
69
2.20
Convolutions of functions in dual
1^(Еп)-ѕрасеѕ
are
continuous
70
2.21
Hilbert-spaces
71
Exercises
75
CHAPTER
3.
Rearrangement Inequalities
79
3.1
Introduction
79
3.2
Definition of functions vanishing at infinity
80
3.3
Rearrangements of sets and functions
80
3.4
The simplest rearrangement inequality
82
3.5
Nonexpansivity of rearrangement
83
3.6
Riesz's rearrangement inequality in one-dimension
84
3.7
Riesz's rearrangement inequality
87
3.8
General rearrangement inequality
93
3.9
Strict rearrangement inequality
93
Exercises
95
CHAPTER
4.
Integral Inequalities
97
4.1
Introduction
97
4.2
Young's inequality
98
4.3
Hardy-Littlewood-Sobolev inequality
106
4.4
Conformai
transformations and stereographic projection
110
4.5
Conformai invariance
of the Hardy-Littlewood-Sobolev
inequality
114
4.6
Competing symmetries
117
4.7
Proof of Theorem
4.3:
Sharp version of the Hardy-
Littlewood-Sobolev inequality
119
4.8
Action of the
conformai
group on optimizers
120
Exercises
121
CHAPTER
5.
The Fourier Transform
125
5.1
Definition of the
Ll
Fourier transform
125
5.2
Fourier transform of a Gaussian
127
5.3
Plancherel's theorem
128
5.4
Definition of the L2 Fourier transform
129
5.5
Inversion formula
130
5.6
The Fourier transform in
Ι^(Εη)
130
5.7
The sharp Hausdorff-Young inequality
131
5.8
Convolutions
132
5.9
Fourier transform of \x\a~n
132
5.10
Extension of
5.9
to IJ>(Wl)
133
Exercises
135
CHAPTER
6.
Distributions
137
6.1
Introduction
137
6.2
Test functions (The space V(U))
138
6.3
Definition of distributions and their convergence
138
6.4
Locally summable functions, Lfoc(Q)
139
6.5
Functions are uniquely determined by distributions
140
6.6
Derivatives of distributions
141
6.7
Definition of WJ£(U) and
Wx*{u)
142
6.8
Interchanging convolutions with distributions
144
6.9
Fundamental theorem of calculus for distributions
145
6.10
Equivalence of classical and distributional derivatives
146
6.11
Distributions with zero derivatives are constants
148
6.12
Multiplication and convolution of distributions by C°°-
functions
148
6.13
Approximation of distributions by C^-functions
149
6.14
Linear dependence of distributions
150
6.15 C°°(ß)
is 'dense' in
W¿f
(Ω)
151
6.16
Chain rule
152
6.17
Derivative of the absolute value
154
6.18
Min
and Max of
W
1)P-functions are in W1*
155
6.19
Gradients vanish on the inverse of small sets
157
6.20
Distributional Laplacian of Green's functions
158
6.21
Solution of
Poisson
's
equation
159
6.22
Positive distributions are measures
161
6.23
Yukawa potential
166
6.24
The dual of W^R")
169
Exercises
170
CHAPTER
7.
The Sobolev Spaces H1 and if1/2
173
7.1
Introduction
173
7.2
Definition of
#Χ(Ω)
173
7.3
Completeness of
ЯХ(П)
174
7.4
Multiplication by functions in
С°°(П)
175
7.5
Remark about
#Χ(Ω)
and
Τ¥1)2(Ω)
176
7.6
Density of
С°°(П)
in
#Τ(Ω)
176
7.7
Partial integration for functions in
Я1^™)
177
7.8
Convexity inequality for gradients
179
7.9
Fourier characterization
of
Я1
(R*1)
181
•
Heat kernel
182
7.10
—Δ
is the infinitesimal generator of the heat kernel
183
7.11
Definition of H^2{Rn)
183
7.12
Integral formulas for
(ƒ,
\p\
ƒ)
and
(ƒ,
y/p2
+
m2
ƒ) 186
7.13
Convexity inequality for the relativistic kinetic
energy
187
7.14
Density of C~(Rn) in
Η^2(Κη)
188
7.15
Action of yj—
Δ
and
ν7—Δ
+
m2 — m on
distributions
188
7.16
Multiplication of tf^-fiinctions by C00-functions
189
7.17
Symmetric decreasing rearrangement decreases kinetic
energy
190
7.18
Weak limits
193
7.19
Magnetic fields: The
Я^
-spaces
193
7.20
Definition of H\(Rn)
194
7.21
Diamagnetic inequality
195
7.22
C^iW1) is dense in H\(Rn)
196
Exercises
197
CHAPTER
8.
Sobolev Inequalities
201
8.1
Introduction
201
8.2
Definition of jD^R") and D1^^")
203
8.3
Sobolev's inequality for gradients
204
8.4
Sobolev's inequality for \p\
206
8.5
Sobolev inequalities in
1
and
2
dimensions
207
8.6
Weak convergence implies strong convergence on small sets
210
8.7
Weak convergence implies a.e. convergence
214
8.8
Sobolev inequalities for Wm^(U)
215
8.9
Rellich-Kondrashov theorem
216
8.10
Nonzero weak convergence after translations
217
8.11
Poincaré's
inequalities for Wm*{u)
220
8.12
Poincaré-Sobolev
inequality for Wm*(u)
221
8.13
Nash's inequality
222
8.14
The logarithmic Sobolev inequality
225
8.15
A glance at contraction semigroups
227
8.16
Equivalence
of Nash's inequality and smoothing estimates
229
8.17
Application to the heat equation
231
8.18
Derivation of the heat kernel via logarithmic Sobolev in¬
equalities
234
Exercises
237
CHAPTER
9.
Potential Theory and Coulomb Energies
239
9.1
Introduction
239
9.2
Definition of harmonic, subharmonic, and superharmonic
functions
240
9.3
Properties of harmonic, subharmonic, and superharmonic
functions
241
9.4
The strong maximum principle
246
9.5
Harnack's inequality
247
9.6
Subharmonic functions are potentials
248
9.7
Spherical charge distributions are 'equivalent' to point
charges
251
9.8
Positivity
properties of the Coulomb energy
252
9.9
Mean value inequality for
Α — μ2
254
9.10
Lower bounds on
Schrödinger
'wave' functions
256
9.11
Unique solution of Yukawa's equation
257
Exercises
258
CHAPTER
10.
Regularity of Solutions of Poisson's
Equation
259
10.1
Introduction
259
10.2
Continuity and first differentiability of solutions of Poisson's
equation
262
10.3
Higher differentiability of solutions of Poisson's equation
264
CHAPTER
11.
Introduction to the Calculus of Variations
269
11.1
Introduction
269
11.2 Schrödinger's
equation
271
11.3
Domination of the potential energy by the kinetic energy
272
11.4
Weak continuity of the potential energy
276
11.5
Existence of a minimizer for Eq
277
11.6 Higher
eigenvalues and eigenfunctions
279
11.7
Regularity of solutions
281
11.8
Uniqueness of minimizers
282
11.9
Uniqueness of positive solutions
283
11.10
The hydrogen atom
284
11.11
The Thomas-Fermi problem
285
11.12
Existence of an unconstrained Thomas-Fermi minimizer
286
11.13
Thomas-Fermi equation
287
11.14
The Thomas-Fermi minimizer
289
11.15
The capacitor problem
291
11.16
Solution of the capacitor problem
295
11.17
Balls have smallest capacity
298
Exercises
299
CHAPTER
12.
More about Eigenvalues
301
12.1
Min-max
principles
302
12.2
Generalized
min-max
304
12.3
Bound for eigenvalue sums in a domain
306
12.4
Bound for
Schrödinger
eigenvalue sums
308
12.5
Kinetic energy with antisymmetry
313
12.6
The semiclassical approximation
316
12.7
Definition of coherent states
318
12.8
Resolution of the identity
319
12.9
Representation of the nonrelativistic kinetic energy
321
12.10
Bounds for the relativistic kinetic energy
321
12.11
Large TV eigenvalue sums in a domain
322
12.12
Large
N
asymptotics of
Schrödinger
eigenvalue sums
325
Exercises
329
List of Symbols
333
References
337
Index
343 |
| any_adam_object | 1 |
| author | Lieb, Elliott H. 1932- Loss, Michael 1954- |
| author_GND | (DE-588)11899655X (DE-588)172513804 |
| author_facet | Lieb, Elliott H. 1932- Loss, Michael 1954- |
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| author_sort | Lieb, Elliott H. 1932- |
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| building | Verbundindex |
| bvnumber | BV037367994 |
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| callnumber-label | QA300 |
| callnumber-raw | QA300 |
| callnumber-search | QA300 |
| callnumber-sort | QA 3300 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 400 SK 600 |
| classification_tum | MAT 260f |
| ctrlnum | (OCoLC)729997514 (DE-599)BVBBV037367994 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition, reprinted with corrections |
| format | Book |
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| id | DE-604.BV037367994 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-09T15:02:16Z |
| institution | BVB |
| isbn | 0821827839 9780821827833 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022521399 |
| oclc_num | 729997514 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-20 DE-703 DE-739 DE-573 DE-91G DE-BY-TUM |
| owner_facet | DE-19 DE-BY-UBM DE-20 DE-703 DE-739 DE-573 DE-91G DE-BY-TUM |
| physical | xxii, 348 Seiten |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Lieb, Elliott H. 1932- Verfasser (DE-588)11899655X aut Analysis Elliott H. Lieb ; Michael Loss Second edition, reprinted with corrections Providence, Rhode Island American Mathematical Society 2010 xxii, 348 Seiten txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 14 Hier auch später erschienene, unveränderte Nachdrucke Analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Loss, Michael 1954- Verfasser (DE-588)172513804 aut Graduate studies in mathematics 14 (DE-604)BV009739289 14 http://www.gbv.de/dms/goettingen/325084793.pdf Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022521399&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lieb, Elliott H. 1932- Loss, Michael 1954- Analysis Graduate studies in mathematics Analysis Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 |
| title | Analysis |
| title_auth | Analysis |
| title_exact_search | Analysis |
| title_full | Analysis Elliott H. Lieb ; Michael Loss |
| title_fullStr | Analysis Elliott H. Lieb ; Michael Loss |
| title_full_unstemmed | Analysis Elliott H. Lieb ; Michael Loss |
| title_short | Analysis |
| title_sort | analysis |
| topic | Analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Analysis |
| url | http://www.gbv.de/dms/goettingen/325084793.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022521399&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT liebelliotth analysis AT lossmichael analysis |
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