Hilbert transforms: 1
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Encyclopedia of mathematics and its applications
124 |
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| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XXXVIII, 858 S. graph. Darst. |
| ISBN: | 9780521887625 |
Internformat
MARC
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| 100 | 1 | |a King, Frederick W. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Hilbert transforms |n 1 |c Frederick W. King |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XXXVIII, 858 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Encyclopedia of mathematics and its applications |v 124 | |
| 490 | 0 | |a Encyclopedia of mathematics and its applications |v ... | |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017539420&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
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Datensatz im Suchindex
| _version_ | 1804139094223093760 |
|---|---|
| adam_text | Contents
Preface
List of symbols
List of abbreviations
page
xxi
xxv
xxxviii
Volume I
Introduction
1.1
Some common integral transforms
1.2
Definition of the Hubert transform
1.3
The Hubert transform as an operator
1.4
Diversity of applications of the Hubert transform
Notes
Exercises
1
1
1
4
6
8
9
2
Review of some background mathematics
2.1
Introduction
2.2
Order symbols O(
)
and o(
)
2.3
Lipschitz and Holder conditions
2.4
Cauchy principal value
2.5
Fourier series
2.5.1
Periodic property
2.5.2
Piecewise continuous functions
2.5.3
Definition of Fourier series
2.5.4
Bessel s inequality
2.6
Fourier transforms
2.6.1
Definition of the Fourier transform
2.6.2
Convolution theorem
2.6.3
The
Parseval
and Plancherel formulas
2.7
The Fourier integral
11
11
11
12
13
14
14
15
16
19
19
19
21
21
22
VII
viii Contents
2.8
Some basic results from complex variable theory
23
2.8.1
Integration of analytic functions
27
2.8.2
Cauchy integral theorem
29
2.8.3
Cauchy integral formula
30
2.8.4
Jordan s lemma
30
2.8.5
The Laurent expansion
31
2.8.6
The Cauchy residue theorem
33
2.8.7
Entire functions
34
2.9
Conformai
mapping
37
2.10
Some functional analysis basics
39
2.10.1
Hubert space
42
2.10.2
The Hardy space HP
43
2.10.3
Topological space
44
2.10.4
Compact operators
45
2.11
Lebesgue measure and integration
45
2.11.1
The notion of measure
48
2.12
Theorems due to Fubini and Tonelli
55
2.13
The
Hardy-Poincaré-Bertrand
formula
57
2.14
Riemann-Lebesgue lemma
61
2.15
Some elements of the theory of distributions
63
2.15.1
Generalized functions as sequences of functions
65
2.15.2
Schwartz distributions
68
2.16
Summation of series: convergence accelerator techniques
70
2.16.1
Richardson extrapolation
71
2.16.2
The Levin sequence transformations
74
Notes
77
Exercises
80
3
Derivation of the
Hubert
transform relations
83
3.1
Hubert transforms
-
basic forms
83
3.2
The
Poisson
integral for the half plane
85
3.3
The
Poisson
integral for the disc
89
3.3.1
The
Poisson
kernel for the disc
91
3.4
Hubert
transform on the real line
94
3.4.1
Conditions on the function
ƒ 96
3.4.2
The
Phragmén-Lindelõf
theorem
100
3.4.3
Some examples
101
3.5
Transformation to other limits
104
3.6
Cauchy integrals
Ю7
3.7
The Plemelj formulas
1
11
3.8
Inversion formula for a Cauchy integral
112
3.9
Hubert transform on the circle
114
Contents
ix
3.10
Alternative approach to the Hubert transform on the circle
115
3.11
Hardy s approach
118
3.11.1
Hubert transform on
R
120
3.12
Fourier integral approach to the Hubert transform on
R
122
3.13
Fourier series approach
129
3.14
The Hubert transform for periodic functions
132
3.15
Cancellation behavior for the Hubert transform
13 5
Notes
141
Exercises
142
4
Some basic properties of the Hubert transform
145
4.1
Introduction
145
4.1.1
Complex conjugation property
145
4.1.2
Linearity
145
4.2
Hubert transforms of even or odd functions
146
4.3
Skew-symmetric character of
Hubert
transform pairs
147
4.4
Inversion property
148
4.5
Scale changes
150
4.5.1
Linear scale changes
150
4.5.2
Some nonlinear scale transformations for the
Hubert
transform
150
4.6
Translation, dilation, and reflection operators
155
4.7
The Hubert transform of the product
x f (x) 1
60
4.8
The Hubert transform of derivatives
164
4.9
Convolution property
167
4.10
Titchmarsh formulas of the
Parseval
type
170
4.11
Unitary property of
Я
174
4.12
Orthogonality property
174
4.13
Hubert transforms via series expansion
177
4.14
The Hubert transform of a product of functions
181
4.15
The Hubert transform product theorem
(Bedrosian s theorem)
184
4.16
A theorem due to
Tricomi
187
4.17
Eigenvalues and eigenftmctions of the Hubert transform operator
195
4.18
Projection operators
199
4.19
A theorem due to Akhiezer
200
4.20
The Riesz inequality
203
4.21
The Hubert transform of functions in L1
andini,00
211
4.21.1
The Incase
214
4.22
Connection between Hubert transforms and causal functions
215
4.23
The
Hardy-Poincaré-Bertrand
formula revisited
223
4.24
A theorem due to McLean and Elliott
226
x
Contents
4.25
The Hilbert-Stieltjes transform
231
4.26
A theorem due to Stein and Weiss
241
Notes
246
Exercises
248
5
Relationship between the Hubert transform
and some common transforms
252
5.1
Introduction
252
5.2
Fourier transform of the Hilbert transform
252
5.3
Even and odd Hilbert transform operators
258
5.4
The commutator
[Τ,Η]
261
5.5
Hartley transform of the Hilbert transform
262
5.6
Relationship between the Hilbert transform and
the
Stieltjes
transform
265
5.7
Relationship between the Laplace transform and
the Hilbert transform
267
5.8
Mellin transform of the Hilbert transform
269
5.9
The Fourier allied integral
275
5.10
The Radon transform
277
Notes
285
Exercises
286
6
The Hilbert transform of periodic functions
288
6.1
Introduction
288
6.2
Approach using infinite product expansions
289
6.3
Fourier series approach
291
6.4
An operator approach to the Hilbert transform on the circle
292
6.5
Hilbert transforms of some standard kernels
296
6.6
The inversion formula
301
6.7
Even and odd periodic functions
303
6.8
Scale changes
304
6.9
Parseval-type formulas
305
6.10
Convolution property
307
6.11
Connection with Fourier transforms
309
6.12
Orthogonality property
310
6.13
Eigenvalues and eigenfunctions of the Hilbert transform operator
311
6.14
Projection operators
312
6.15
The
Hardy-Poincaré-Bertrand
formula
313
6.16
A theorem due to
Prívalov
3
1
6
6.17
The Marcel Riesz inequality
318
6.18
The partial sum of a Fourier series
323
6.19
Lusin s conjecture
325
Notes
328
Exercises
329
Contents xi
7
Inequalities for the Hubert
transform
331
7.1
The Marcel Riesz inequality revisited
331
7.1.1
Hubert s integral
335
7.2
A Kolmogorov inequality
335
7.3
A Zygmund inequality
340
7.4
A Bernstein inequality
343
7.5
The Hubert transform of a function having a bounded
integral and derivative
350
7.6
Connections between the Hubert transform on
R
and
T
352
7.7
Weighted norm inequalities for the Hubert transform
354
7.8
Weak-type inequalities
364
7.9
The Hardy-Littlewood maximal function
368
7.10
The maximal Hubert transform function
373
7.11
A theorem due to Helson and
Szegő
386
7.12
The Ap condition
388
7.13
A theorem due to Hunt, Muckenhoupt, and Wheeden
395
7.13.1
Weighted norm inequalities for #e and Ho
399
7.14
Weighted norm inequalities for the Hubert transform
of functions with vanishing moments
402
7.15
Weighted norm inequalities for the Hubert transform
with two weights
403
7.16
Some miscellaneous inequalities for the Hubert transform
408
Notes
413
Exercises
416
8
Asymptotic behavior of the
Hubert
transform
419
8.1
Asymptotic expansions
419
8.2
Asymptotic expansion of the
Stieltjes
transform
420
8.3
Asymptotic expansion of the one-sided
Hubert
transform
422
Notes
434
Exercises
434
9
Hubert transforms of some special functions
438
9.1
Hubert transforms of special functions
438
9.2
Hubert transforms involving Legendre polynomials
438
9.3
Hubert transforms of the Hermite polynomials
with a Gaussian weight
446
9.4
Hubert transforms of the Laguerre polynomials with
a weight function H(x)e~x
449
9.5
Other orthogonal polynomials
452
9.6
Bessel functions of the first kind
453
9.7
Bessel functions of the first and second kind
for non-integer index
459
xii
Contents
9.8
The
Strave
function
460
9.9
Spherical Bessel functions
462
9.10
Modified Bessel functions of the first and second kind
465
9.11
The cosine and sine integral functions
470
9.12
The Weber and Anger functions
470
Notes
471
Exercises
472
10
Hubert
transforms involving distributions
474
10.1
Some basic distributions
474
10.2
Some important spaces for distributions
478
10.3
Some key distributions
483
10.4
The Fourier transform of some key distributions
487
10.5
A Parseval-type formula approach to
HT
490
10.6
Convolution operation for distributions
491
10.7
Convolution and the Hubert transform
495
10.8
Analytic representation of distributions
498
10.9
The inversion formula
502
10.10
The derivative property
504
10.11
The Fourier transform connection
505
10.12
Periodic distributions: some preliminary notions
508
10.13
The Hubert transform of periodic distributions
515
10.14
The Hubert transform of
ultradistributions
and related ideas
516
Notes
518
Exercises
519
521
521
523
527
527
529
536
541
543
544
544
546
550
11
The
finite Hubert transform
11.1
Introduction
11.2
Alternative formulas: the cosine form
11.2.1
A result due to Hardy
11.3
The cotangent form
11.4
The inversion formula:
Tricomi
s
approach
11.4.1
Inversion of the finite Hubert transform for
the interval
(0,1)
11.5
Inversion by a Fourier series approach
11.6
The Riemann problem
11.7
The Hubert problem
11.8
The Riemann-Hilbert problem
11.8.1
The index of a function
11.9
Carleman s approach
Contents xiii
11.10
Some basic properties of the finite Hubert transform
552
11.10.1
Even-odd character
552
11.10.2
Inversion property
552
11.10.3
Scale changes
553
11.10.4
Finite
Hubert
transform of the product xnf{x)
554
11.10.5
Derivative of the finite Hubert transform
555
11.10.6
Convolution property
556
11.10.7
Fourier transform of the finite Hubert transform
557
11.10.8
Parseval-type identities
559
11.10.9
Orthogonality property
562
11.10.10
Eigenfiinctions and eigenvalues of the finite
Hubert transform operator
563
11.11
Finite Hubert transform of the Legendre polynomials
563
11.12
Finite Hubert transform of the Chebyshev polynomials
567
11.13
Contour integration approach to the derivation of some finite
Hubert transforms
570
11.14
The thin airfoil problem
577
11.15
The generalized airfoil problem
582
11.16
The cofmite Hubert transform
582
Notes
584
Exercises
585
12
Some singular integral equations
588
12.1
Introduction
588
12.2
Fredholm
equations of the first kind
589
12.3
Fredholm
equations of the second kind
592
12.4
Fredholm
equations of the third kind
594
12.5
Fourier transform approach to solving singular integral equations
599
12.6
A finite Hubert transform integral equation
601
12.7
The one-sided Hubert transform
609
12.7.1
Eigenfimctions and eigenvalues of the one-sided
Hubert transform operator
612
12.8
Fourier transform approach to the inversion of the one-sided
Hubert transform
612
12.9
An inhomogeneous singular integral equation for H
614
12.10
A nonlinear singular integral equation
617
12.11
The Peierls-Nabarro equation
618
12.12
The sine-Hilbert equation
620
12.13
The Benjamin-Ono equation
624
12.13.1
Conservation laws
627
12.14
Singular integral equations involving distributions
630
Notes
632
Exercises
633
xiv
Contents
13
Discrete Hilbert transforms
637
13.1
Introduction
637
13.2
The discrete Fourier transform
637
13.3
Some properties of the discrete Fourier transform
640
13.4
Evaluation of the DFT
641
13.5
Relationship between the DFT and the Fourier transform
643
13.6
The
Z
transform
644
13.7
Z
transform of a product
647
13.8
The Hilbert transform of a discrete time signal
649
13.9
Z
transform of a causal sequence
652
13.10
Fourier transform of a causal sequence
656
13.11
The discrete Hilbert transform in analysis
660
13.12
Hubert s inequality
661
13.13
Alternative approach to the discrete Hilbert transform
666
13.14
Discrete analytic functions
675
13.15
Weighted discrete Hilbert transform inequalities
679
Notes
680
Exercises
681
14
Numerical evaluation of Hilbert transforms
684
14.1
Introduction
684
14.2
Some elementary transformations for Cauchy
principal value integrals
684
14.3
Some classical formulas for numerical quadrature
688
14.3.1
A Maclaurin-type formula
688
14.3.2
The trapezoidal rule
690
14.3.3
Simpson s rule
691
14.4
Gaussian quadrature: some basics
691
14.5
Gaussian quadrature: implementation procedures
694
14.6
Specialized Gaussian quadrature: application to the
Hilbert transform
701
14.6.1
Error estimates
706
14.7
Specialized Gaussian quadrature: application to He and Ho
708
14.8
Numerical integration of the Fourier transform
709
14.9
The fast Fourier transform: numerical implementation
7
1
1
14.10
Hubert transform via the fast Fourier transform
712
14.11
The Hilbert transform via the allied Fourier integral
712
14.12
The Hilbert transform via conjugate Fourier series
713
14.13
The Hilbert transform of oscillatory functions
717
14.14
An eigenfunction expansion
723
14.15
The finite Hilbert transform
727
Contents xv
Notes 730
Exercises
732
Appendix 14.1 735
Appendix 14.2 744
References
745
Author index
824
Subject index
840
HiLBERT
Transforms
The Hubert transform arises widely in a variety of applications, including problems
in aerodynamics, condensed matter physics, optics, fluids, and engineering.
This work, written in an easy-to-use style, is destined to become the definitive
reference on the subject. It contains a thorough discussion of all the common Hubert
transforms, mathematical techniques for evaluating them, and a detailed discussion
of their application. Especially valuable features are the tabulation of analytically
evaluated Hubert transforms, and an atlas that immediately illustrates how the
Hubert transform alters a function. These will provide useful and convenient
resources for researchers.
A collection of exercises is provided for the reader to test comprehension of the
material in each chapter. The bibliography is an extensive collection of references to
both the classical mathematical papers, and to a diverse array of applications.
Frederick W. King is a Professor in the Department of Chemistry at the University
of Wisconsin-Eau Claire.
|
| any_adam_object | 1 |
| author | King, Frederick W. |
| author_facet | King, Frederick W. |
| author_role | aut |
| author_sort | King, Frederick W. |
| author_variant | f w k fw fwk |
| building | Verbundindex |
| bvnumber | BV035482905 |
| classification_rvk | SK 450 |
| ctrlnum | (OCoLC)634244978 (DE-599)BVBBV035482905 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV035482905 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:38:37Z |
| institution | BVB |
| isbn | 9780521887625 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017539420 |
| oclc_num | 634244978 |
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| owner | DE-91G DE-BY-TUM DE-634 DE-83 DE-19 DE-BY-UBM DE-11 DE-703 DE-384 DE-824 |
| owner_facet | DE-91G DE-BY-TUM DE-634 DE-83 DE-19 DE-BY-UBM DE-11 DE-703 DE-384 DE-824 |
| physical | XXXVIII, 858 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | King, Frederick W. Verfasser aut Hilbert transforms 1 Frederick W. King 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2009 XXXVIII, 858 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 124 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 1 Encyclopedia of mathematics and its applications 124 (DE-604)BV000903719 124 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017539420&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017539420&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| 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 1 Frederick W. King |
| title_fullStr | Hilbert transforms 1 Frederick W. King |
| title_full_unstemmed | Hilbert transforms 1 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=017539420&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=017539420&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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