A basis theory primer:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Dordrecht ; Heidelberg ; London
Birkhäuser
[2011]
|
| Edition: | Expanded edition |
| Series: | Applied and numerical harmonic analysis
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverzeichnis Seite [515] - 526 |
| Physical Description: | xxv, 534 Seiten Diagramme |
| ISBN: | 9780817646868 |
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MARC
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Record in the Search Index
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| adam_text | The classical subject of bases in Banach spaces has taken on a new life in the modem
development of applied harmonic analysis. This textbook is a self-contained introduction
to the abstract theory of bases and redundant frame expansions and its use in both applied
and classical harmonic analysis.
The four parts of the text take the reader from classical functional analysis and
basis theory to modern time-frequency and wavelet theory.
•
Part I develops the functional analysis that underlies most of the concepts
presented in the later parts of the text.
•
Part II presents the abstract theory of bases and frames in Banach and Hubert
spaces, including the classical topics of convergence,
Schauder
bases,
biorthogonal
systems, and unconditional bases, followed by the more recent topics of Riesz bases
and frames in Hubert spaces.
•
Part III relates bases and frames to applied harmonic analysis, including sampling
theory,
Gabor
analysis, and wavelet theory.
•
Part IV deals with classical harmonic analysis and Fourier series, emphasizing
the role played by bases, which is a different viewpoint from that taken in most
discussions of Fourier series.
Key features:
•
Self-contained presentation with clear proofs is accessible to graduate students,
pure and applied mathematicians, and engineers interested in the mathematical
underpinnings of applications.
•
Extensive exercises complement the text and provide opportunities for learning-
by-doing, making the text suitable for graduate-level courses; hints for selected
exercises are included at the end of the book.
•
A separate solutions manual is available for instructors upon request at
No other text develops the ties between classical basis theory and its modern uses
in applied harmonic analysis.
Contents
ANHA
Series
Preface
..........................................
vii
Preface
........................................................ xv
General
Notation.............................................xxiii
Part I A Primer on Functional Analysis
1
Banach Spaces and Operator Theory
....................... 3
1.1
Definition and Examples of Banach Spaces
.................. 3
1.2
Holder s and Minkowski s Inequalities
...................... 9
1.3
Basic Properties of Banach Spaces
......................... 13
1.4
Linear Combinations, Sequences, Series, and Complete Sets
... 20
1.5 Hubert
Spaces
........................................... 25
1.6
Orthogonal Sequences in Hubert Spaces
.................... 32
1.7
Operators
............................................... 42
1.8
Bounded Linear Functionals and the Dual Space
............. 50
2
Functional Analysis
........................................ 57
2.1
The Hahn-Banach Theorem and Its Implications
............ 57
2.2
Reflexivity
.............................................. 61
2.3
Adjoints
of Operators on Banach Spaces
.................... 62
2.4
Adjoints
of Operators on Hubert Spaces
.................... 64
2.5
The Baire Category Theorem
.............................. 69
2.6
The Uniform Boundedness Principle
........................ 70
2.7
The Open Mapping Theorem
.............................. 73
2.8
Topologica!
Isomorphisms
................................. 75
2.9
The Closed Graph Theorem
............................... 78
2.10
Weak Convergence
....................................... 79
Part II Bases and Frames
3
Unconditional Convergence of Series in Banach and
Hubert Spaces
............................................. 87
3.1
Convergence, Absolute Convergence, and Unconditional
Convergence of Series
..................................... 88
3.2
Convergence with Respect to the Directed Set of Finite
Subsets of
N............................................ 92
3.3
Equivalent Characterizations of Unconditional Convergence
... 94
3.4
Further Results on Unconditional Convergence
..............100
3.5
Unconditional Convergence of Series in Hubert Spaces
........104
3.6
The Dvoretzky-Rogers Theorem
...........................116
4
Bases in Banach Spaces
....................................125
4.1
Hamel
Bases
............................................125
4.2
Bases
...................................................128
4.3 Schauder
Bases
..........................................130
4.4
Equivalent Bases
.........................................140
4.5
Schauder s Basis for C[0,
1]................................142
4.6
The Trigonometric System
................................144
4.7
Weak and Weak* Bases in Banach Spaces
...................147
5
Biorthogonality, Minimality, and More About Bases
.......153
5.1
The Connection between Minimality and Biorthogonality
.....153
5.2
Shades of Grey: Independence
.............................156
5.3
A Characterization of
Schauder
Bases
......................161
5.4
A Characterization of Minimal Sequences and
Schauder
Bases
. 165
5.5
The
Haar
System in Lp[0,
1]...............................168
5.6
Duality for Bases
........................................170
5.7
Perturbations of Bases
....................................172
6
Unconditional Bases in Banach Spaces
.....................177
6.1
Basic Properties and the Unconditional Basis Constant
.......177
6.2
Characterizations of Unconditional Bases
...................180
6.3
Conditionality of the
Schauder
System in C[0,
1].............184
6.4
Conditionality of the
Haar
System in
Ьх[0,
1]................187
Bessel Sequences and Bases in Hubert Spaces
.............. 189
7.1
Bessel Sequences in Hubert Spaces
.........................189
7.2
Unconditional Bases and Riesz Bases in Hubert Spaces
.......195
8
Frames
in Hubert Spaces
...................................203
8.1
Definition and Motivation
.................................204
8.2
Frame Expansions and the Frame Operator
.................214
8.3
Overcompleteness
........................................221
8.4
Frames and Bases
........................................224
8.5
Characterizations of Frames
...............................226
8.6
Convergence of Frame Series
..............................233
8.7
Excess
..................................................238
Part III Bases and Frames in Applied Harmonic Analysis
9
The Fourier Transform on the Real Line
...................249
9.1
Summary: Main Properties of the Fourier Transform on the
Real Line
...............................................250
9.2
Motivation: The Trigonometric System
.....................251
9.3
The Fourier Transform on LX(R)
..........................253
9.4
The Fourier Transform on L2(R)
..........................262
9.5
Absolute Continuity
......................................265
10
Sampling, Weighted Exponentials, and Translations
........267
10.1
Bandlimited Functions
....................................269
10.2
The Sampling Theorem
...................................274
10.3
Frames of Weighted Exponentials
..........................277
10.4
Frames of Translates
............ . ........................282
11 Gabor
Bases and Frames
...................................301
11.1
Time-Frequency Shifts
....................................302
11.2
Painless
Nonorthogonal
Expansions
........................306
11.3
The Nyquist Density and Necessary Conditions for Frame
Bounds
.................................................311
11.4
Wiener Amalgam Spaces
..................................315
11.5
The Walnut Representation
...............................319
11.6
The
Zak
Transform
......................................324
11.7 Gabor
Systems at the Critical Density
......................331
11.8
The Balian-Low Theorem
.................................335
11.9
The
HRT
Conjecture
.....................................341
12
Wavelet Bases and Frames
.................................351
12.1
Some Basic Facts
........................................352
12.2
Wavelet Frames and Wavelet Sets
..........................354
12.3
Frame Bounds and the Admissibility Condition
..............362
12.4
Multiresolution Analysis
..................................370
12.5
All About the Scaling Function, I: Refinability
...............378
12.6
All About the Scaling Function, II: Existence
................393
xiv Contents
12.7 All
About the Wavelet
....................................406
12.8
Examples
...............................................417
Part IV Fourier Series
13
Fourier Series
..............................................429
13.1
Notation and Terminology
................................429
13.2
Fourier Coefficients and Fourier Series
......................431
13.3
Convolution
.............................................434
13.4
Approximate Identities
...................................439
13.5
Partial Sums and the Dirichlet Kernel
......................443
13.6
Cesàro
Summability and the
Fejér
Kernel
...................447
13.7
The Inversion Formula for LX(T)
..........................452
14
Basis Properties of Fourier Series
..........................455
14.1
The Partial Sum Operators
...............................455
14.2
The Conjugate Function
..................................460
14.3
Pointwise Almost Everywhere Convergence
..................464
Part V Appendices
A Lebesgue Measure and Integration
.........................469
A.I Exterior Lebesgue Measure
.......
v
........................469
A.2 Lebesgue Measure
.......................................470
A.3 Measurable Functions
....................................472
A.4 The Lebesgue Integral
....................................473
A.
5
Lp Spaces and Convergence
...............................475
A.6 Repeated Integration
.....................................477
В
Compact and Hilbert-Schmidt Operators
..................481
B.I Compact Sets
...........................................481
B.2 Compact Operators
......................................482
B.3 Hilbert-Schmidt Operators
................................484
B.4 Finite-Rank Operators and Tensor Products
.................485
B.5 The Hilbert-Schmidt Kernel Theorem
......................488
Hints for Exercises
............................................491
Index of Symbols
..............................................511
References
.....................................................515
Index
..........................................................527
|
| any_adam_object | 1 |
| author | Heil, Christopher 1960- |
| author_GND | (DE-588)132081466 |
| author_facet | Heil, Christopher 1960- |
| author_role | aut |
| author_sort | Heil, Christopher 1960- |
| author_variant | c h ch |
| building | Verbundindex |
| bvnumber | BV037368896 |
| classification_rvk | SK 600 |
| ctrlnum | (OCoLC)707190780 (DE-599)DNB98348709X |
| discipline | Mathematik |
| edition | Expanded edition |
| format | Book |
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| id | DE-604.BV037368896 |
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| indexdate | 2024-07-09T23:22:48Z |
| institution | BVB |
| isbn | 9780817646868 |
| language | English |
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| physical | xxv, 534 Seiten Diagramme |
| publishDate | 2011 |
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| publishDateSort | 2011 |
| publisher | Birkhäuser |
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| series2 | Applied and numerical harmonic analysis |
| spelling | Heil, Christopher 1960- Verfasser (DE-588)132081466 aut A basis theory primer Christopher Heil Expanded edition New York ; Dordrecht ; Heidelberg ; London Birkhäuser [2011] © 2011 xxv, 534 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Applied and numerical harmonic analysis Literaturverzeichnis Seite [515] - 526 Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Funktionalanalysis (DE-588)4018916-8 s DE-604 Erscheint auch als Online-Ausgabe 978-0-8176-4687-5 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=022522279&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA 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=022522279&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Heil, Christopher 1960- A basis theory primer Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4123623-3 |
| title | A basis theory primer |
| title_auth | A basis theory primer |
| title_exact_search | A basis theory primer |
| title_full | A basis theory primer Christopher Heil |
| title_fullStr | A basis theory primer Christopher Heil |
| title_full_unstemmed | A basis theory primer Christopher Heil |
| title_short | A basis theory primer |
| title_sort | a basis theory primer |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Funktionalanalysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022522279&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022522279&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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