A handbook of Fourier theorems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge u.a.
Cambridge Univ. Pr.
1989
|
| Edition: | 1. paperback ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 185 S. |
| ISBN: | 0521265037 0521366887 |
Staff View
MARC
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| 245 | 1 | 0 | |a A handbook of Fourier theorems |c D. C. Champeney |
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| 264 | 1 | |a Cambridge u.a. |b Cambridge Univ. Pr. |c 1989 | |
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Record in the Search Index
| _version_ | 1804118026230956032 |
|---|---|
| adam_text | CONTENTS
Preface
xi
1
Introduction
1
2
Lebesgue integration
4
2.1
Introduction
4
2.2
Riemann integration
4
2.3
Null sets
5
2.4
The Lebesgue integral
6
2.5
Nomenclature
8
2.6
Conditions for integrability; measurability
9
2.7
Functions in Lp
12
2.8
Integrals in several dimensions
13
2.9
Alternative approaches
14
3
Some useful theorems
15
3.1
The Minkowski inequality
15
3.2
Holder s theorem
16
3.3
Young s theorem
17
3.4
The Fubini and Tonelli theorems
18
3.5
Two theorems of Lebesgue
19
3.6
Absolute and uniform continuity
20
3.7
The Riemann-Lebesgue theorem
23
4
Convergence of sequences of functions
24
4.1
Introduction
24
4.2
Pointwise convergence
24
4.3
Bounded, dominated and monotone convergence
25
4.4
Uniform convergence
26
4.5
Convergence in the mean
27
4.6
Cauchy sequences
29
viii Contents
5
Local
averages and convolution kernels
30
5.1
Introduction
30
5.2
Lebesgue points
31
5.3
Approximate convolution identities
33
5.4
The Dirichlet kernel and Dirichlet points
35
5.5
The functions of
du
Bois-Reymond and of
Fejér
36
5.6
Carleson
s
theorem
37
5.7
KolmogorofPs theorem
37
5.8
The Dirichlet conditions
38
5.9
Jordan s theorem
39
5.10
Dini s theorem
41
5.11
The
de la Vallée-Poussin
test
43
6
Some general remarks on Fourier transformation
44
6.1
Introduction
44
6.2
The definition of the Fourier transform
44
6.3
Sufficient conditions for transformability
47
6.4
Necessary conditions for transformability
49
7
Fourier theorems for good functions
51
7.1
Introduction
51
7.2
Inversion, differentiation and convolution theorems
52
7.3
Good functions of bounded support
55
8
Fourier theorems in V
60
8.1
Basic theorems and definitions
60
8.2
More inversion theorems in
W
63
8.3
Convolution and product theorems in IP
71
8.4
Uncertainty principle and bandwidth theorem
75
8.5
The sampling theorem
77
8.6
Hubert transforms and causal functions
78
9
Fourier theorems for functions outside Lp
81
9.1
Introduction
81
9.2
Functions in class
К
82
9.3
Convolutions and products in
К
85
9.4
Functions outside
К
87
10
Miscellaneous theorems
91
10.1
Differentiation and integration
91
10.2
The Gibbs phenomenon
93
10.3
Complex Fourier transforms
95
10.4
Positive-definite and distribution functions
99
11
Power spectra and Wiener s theorems
102
11.1
Introduction
102
11.2
The autocorrelation function
104
Contents ix
11.3
The spectrum and spectral density
106
11.4
Discrete spectra
109
11.5
Continuous spectra
113
11.6
Miscellaneous theorems
114
12
Generalized functions
118
12.1
Introduction
118
12.2
The definition of functionals in S
119
12.3
Basic theorems
123
12.4
Examples of generalized functions
127
13
Fourier transformation of generalized functions I
135
13.1
Definition of the transform
135
13.2
Simple properties of the transform
136
13.3
Examples of Fourier transforms
137
13.4
The convolution and product of functionals
139
14
Fourier transformation of generalized functions II
145
14.1
Functionals of types D and Z
145
14.2
Fourier transformation of functionals in D
149
14.3
Transformation of products and convolutions in D
152
15
Fourier series
155
15.1
Fourier coefficients of a periodic function
155
15.2
The convergence of Fourier series
156
15.3
Summability of Fourier series
158
15.4
Mean convergence of Fourier series
159
15.5
Sampling theorems
162
15.6
Differentiation and integration of Fourier series
164
15.7
Products and convolutions
166
16
Generalized Fourier series
170
16.1
Introduction
170
16.2
Generalized Fourier coefficients
171
16.3
The Fourier formulae
172
16.4
Differentiation, repetition and sampling
173
16.5
Products and convolutions
175
Bibliography
177
Index
181
|
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| dewey-ones | 515 - Analysis |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 1. paperback ed. |
| format | Book |
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| id | DE-604.BV003682430 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:03:45Z |
| institution | BVB |
| isbn | 0521265037 0521366887 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002342861 |
| oclc_num | 30862344 |
| open_access_boolean | |
| owner | DE-384 DE-188 DE-739 |
| owner_facet | DE-384 DE-188 DE-739 |
| physical | IX, 185 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | Cambridge Univ. Pr. |
| record_format | marc |
| spelling | Champeney, D. C. Verfasser (DE-588)172018021 aut A handbook of Fourier theorems D. C. Champeney 1. paperback ed. Cambridge u.a. Cambridge Univ. Pr. 1989 IX, 185 S. txt rdacontent n rdamedia nc rdacarrier Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Fourier-Reihe (DE-588)4155109-6 gnd rswk-swf Fourier-Reihe (DE-588)4155109-6 s DE-604 Harmonische Analyse (DE-588)4023453-8 s 1\p DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002342861&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Champeney, D. C. A handbook of Fourier theorems Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4155109-6 |
| title | A handbook of Fourier theorems |
| title_auth | A handbook of Fourier theorems |
| title_exact_search | A handbook of Fourier theorems |
| title_full | A handbook of Fourier theorems D. C. Champeney |
| title_fullStr | A handbook of Fourier theorems D. C. Champeney |
| title_full_unstemmed | A handbook of Fourier theorems D. C. Champeney |
| title_short | A handbook of Fourier theorems |
| title_sort | a handbook of fourier theorems |
| topic | Fourier analysis Harmonische Analyse (DE-588)4023453-8 gnd Fourier-Reihe (DE-588)4155109-6 gnd |
| topic_facet | Fourier analysis Harmonische Analyse Fourier-Reihe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002342861&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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