Hilbert transforms: 2
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXXVIII, 660 S. graph. Darst. |
| ISBN: | 9780521517201 0521517206 |
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MARC
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| 100 | 1 | |a King, Frederick W. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Hilbert transforms |n 2 |c Frederick W. King |
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Record in the Search Index
| _version_ | 1804139403687231488 |
|---|---|
| adam_text | Contents
Preface
page
xxi
List of symbols
xxv
List of abbreviations
xxxviii
Volume II
15
Hilbert transforms in E
1
15.1
Definition of the Hilbert transform in En
1
15.2
Definition of the «-dimensional Hilbert transform
5
15.3
The double Hilbert transform
8
15.4
Inversion property for the «-dimensional Hilbert transform
10
15.5
Derivative of the
л
-dimensional Hilbert transform
11
15.6
Fourier transform of the
я
-dimensional Hilbert transform
12
15.7
Relationship between the «-dimensional Hilbert transform and
translation and dilation operators
14
15.8
The Parseval-type formula
16
15.9
Eigenvalues and eigenfunctions of the «-dimensional
Hilbert transform
17
15.10
Periodic functions
18
15.11
A Calderón-Zygmund
inequality
21
15.12
The Riesz transform
25
15.13
The
л
-dimensional Hilbert transform of distributions
32
15.14
Connection with analytic functions
38
Notes
41
Exercises
42
16
Some further extensions of the classical Hilbert transform
44
16.1
Introduction
44
16.2
An extension due to Redheffer
44
16.3
Kober s definition for the L°° case
47
vii
viii Contents
16.4
The
Boas
transform 49
16.4.1
Connection with the Hubert transform
49
16.4.2
Parseval-type formula for the Boas transform
51
16.4.3
Iteration formula for the Boas transform
52
1
6.4.4
Riesz-type bound for the Boas transform
52
16.4.5
Fourier transform of the Boas transform
53
16.4.6
Two theorems due to Boas
54
16.4.7
Inversion of the Boas transform
55
16.4.8
Generalization of the Boas transform
56
16.5
The bilinear Hubert transform
58
16.6
The
vectorial
Hubert transform
60
16.7
The directional Hubert transform
60
16.8
Hubert transforms along curves
62
16.9
The ergodic Hilbert transform
63
16.10
The helical Hilbert transform
66
16.11
Some miscellaneous extensions of the Hilbert transform
67
Notes
69
Exercises
70
17
Linear systems and causality
73
17.1
Systems
73
17.2
Linear systems
73
17.3
Sequential systems
79
17.4
Stationary systems
79
17.5
Primitive statement of causality
80
17.6
The frequency domain
81
17.7
Connection to analyticity
83
17.7.1
A generalized response function
87
17.8
Application of a theorem due to Titchmarsh
90
17.9
An acausal example
93
17.10
The Paley-Wiener log-integral theorem
95
17.11
Extensions of the causality concept
102
17.12
Basic quantum scattering: causality conditions
105
17.13
Extension of Titchmarsh s theorem for distributions
110
Notes
П6
Exercises
117
18
The Hilbert transform of waveforms
and signal processing
119
18.1
Introductory ideas on signal processing
119
18.2
The Hilbert filter
121
18.3
The auto-convolution, cross-correlation, and
auto-correlation functions
123
Contents ix
18.4
The analytic
signal
126
18.5
Amplitude modulation
135
18.6
The frequency domain
138
18.7
Some useful step and pulse functions
139
18.7.1
The Heaviside function
139
18.7.2
The
signum
function
142
18.7.3
The rectangular pulse function
143
18.7.4
The triangular pulse function
145
18.7.5
The sine pulse function
145
18.8
The Hubert transform of step functions and pulse forms
146
18.9
The fractional Hubert transform: the
Lohmann-Mendlovic-Zalevsky definition
147
18.10
The fractional Fourier transform
149
18.11
The fractional Hubert transform: Zayed s definition
159
18.12
The fractional Hubert transform: the Cusmariu definition
160
18.13
The discrete fractional Fourier transform
163
18.14
The discrete fractional Hubert transform
168
18.15
The fractional analytic signal
169
18.16
Empirical mode decomposition: the Hilbert-Huang transform
170
Notes
178
Exercises
180
19
Kramers-Kronig relations
182
19.1
Some background from classical electrodynamics
182
19.2
Kramers-Kronig relations: a simple derivation
184
19.3
Kramers-Kronig relations: a more rigorous derivation
190
19.4
An alternative approach to the Kramers-Kronig relations
197
19.5
Direct derivation of the Kramers-Kronig
relations on the interval
[0,
oo)
199
19.6
The refractive index: Kramers-Kronig relations
201
19.7
Application of Herglotz functions
208
19.8
Conducting materials
216
19.9
Asymptotic behavior of the dispersion relations
219
19.10
Sum rales for the dielectric constant
222
19.11
Sum rales for the refractive index
227
19.12
Application of some properties of the Hubert transform
231
19.13
Sum rales involving weight functions
236
19.14
Summary of sum rales for the dielectric constant
and refractive index
239
19.15
Light scattering: the forward scattering amplitude
239
Notes
247
Exercises
250
x
Contents
20 Dispersion
relations for some linear optical properties
252
20.1
Introduction
252
20.2
Dispersion relations for the normal-incident
reflectance and phase
252
20.3
Sum rules for the reflectance and phase
263
20.4
The conductance: dispersion relations
267
20.5
The energy loss function: dispersion relations
269
20.6
The permeability: dispersion relations
271
20.7
The surface impedance: dispersion relations
274
20.8 Anisotropie
media
278
20.9
Spatial dispersion
280
20.10
Fourier series representation
290
20.11
Fourier series approach to the reflectance
294
20.12
Fourier and allied integral representation
298
20.13
Integral inequalities
300
Notes
303
Exercises
304
21
Dispersion relations for magneto-optical and natural optical activity
306
21.1
Introduction
306
21.2
Circular polarization
307
21.3
The complex refractive indices N+ and N-
309
21.4
Are there dispersion relations for the individual
complex refractive indices N+ and
AL.?
316
21.5
Magnetic optical activity: Faraday effect and
magnetic circular dichroism
319
21.6
Sum rules for magneto-optical activity
323
21.7
Magnetoreflectivity
325
21.8
Optical activity
330
21.9
Dispersion relations for optical activity
345
21.10
Sum rales for optical activity
346
Notes
348
Exercises
349
22
Dispersion relations for nonlinear optical properties
351
22.1
Introduction
351
22.2
Some types of nonlinear optical response
357
22.3
Classical description: the anharmonic oscillator
359
22.4
Density matrix treatment
362
22.5
Asymptotic behavior for the nonlinear susceptibility
372
22.6
One-variable dispersion relations for the nonlinear
susceptibility
377
Contents xi
22.7
Experimental
verification of the dispersion relations for the
nonlinear susceptibility
384
22.8
Dispersion relations in two variables
386
22.9
и
-dimensional
dispersion relations
387
22.10
Situations where the dispersion relations do not hold
388
22.11
Sum rules for the nonlinear susceptibilities
392
22.12
Summary of sum rules for the nonlinear susceptibilities
395
22.13
The nonlinear refractive index and the nonlinear permittivity
395
Notes
403
Exercises
404
23
Some further applications of Hubert transforms
406
23.1
Introduction
406
23.2
Hubert transform spectroscopy
406
23.2.1
The
Josephson
junction
406
23.2.2
Absorption enhancement
410
23.3
The phase retrieval problem
4
11
23.4
X-ray crystallography
417
23.5
Electron-atom scattering
422
23.5.1
Potential scattering
422
23.5.2
Dispersion relations for potential scattering
425
23.5.3
Dispersion relations for electron-H atom scattering
428
23.6
Magnetic resonance applications
433
23.7
DISPA
analysis
435
23.8
Electrical circuit analysis
437
23.9
Applications in acoustics
444
23.10
Viscoelastic behavior
447
23.11
Epilog
448
Notes
449
Exercises
451
Appendix
1
Table of selected Hilbert transforms
453
Appendix
2
Atlas of selected Hilbert transform pairs
534
References
547
Author index
626
Subject index
642
|
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| isbn | 9780521517201 0521517206 |
| language | English |
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| physical | XXXVIII, 660 S. graph. Darst. |
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| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | King, Frederick W. Verfasser aut Hilbert transforms 2 Frederick W. King 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2009 XXXVIII, 660 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 125 Encyclopedia of mathematics and its applications ... Hilbert-Transformation (DE-588)4375311-5 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Hilbert-Transformation (DE-588)4375311-5 s DE-604 (DE-604)BV035482810 2 Encyclopedia of mathematics and its applications 125 (DE-604)BV000903719 125 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017745910&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | King, Frederick W. Hilbert transforms Encyclopedia of mathematics and its applications Hilbert-Transformation (DE-588)4375311-5 gnd |
| subject_GND | (DE-588)4375311-5 (DE-588)4123623-3 |
| title | Hilbert transforms |
| title_auth | Hilbert transforms |
| title_exact_search | Hilbert transforms |
| title_full | Hilbert transforms 2 Frederick W. King |
| title_fullStr | Hilbert transforms 2 Frederick W. King |
| title_full_unstemmed | Hilbert transforms 2 Frederick W. King |
| title_short | Hilbert transforms |
| title_sort | hilbert transforms |
| topic | Hilbert-Transformation (DE-588)4375311-5 gnd |
| topic_facet | Hilbert-Transformation Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017745910&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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