The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wide-ranging collection of references both to the classical mathematical papers, and to a diverse array of applications.
Author(s): Frederick W. King
Series: Encyclopedia of Mathematics and its Applications
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 696
Tags: Математика;Операционное исчисление;
About......Page 2
Encyclopedia of Mathematics and its Applications 125......Page 3
Hilbert transforms Volume 2......Page 4
0521517206......Page 5
Volume II......Page 8
Volume I......Page 13
Preface......Page 22
Symbols......Page 26
15.1 Definition of the Hilbert transform in E_n......Page 41
15.2 Definition of the n-dimensional Hilbert transform ......Page 45
15.3 The double Hilbert transform ......Page 48
15.4 Inversion property for the n-dimensional Hilbert transform ......Page 50
15.5 Derivative of the n-dimensional Hilbert transform ......Page 51
15.6 Fourier transform of the n-dimensional Hilbert transform ......Page 52
15.7 Relationship between the n-dimensional Hilbert transform and translation and dilation operators ......Page 54
15.8 The Parseval-type formula ......Page 56
15.9 Eigenvalues and eigenfunctions of the n-dimensional Hilbert transform ......Page 57
15.10 Periodic functions ......Page 58
15.11 A Calderón-Zygmund inequality ......Page 61
15.12 The Riesz transform ......Page 65
15.13 The n-dimensional Hilbert transform of distributions ......Page 72
15.14 Connection with analytic functions ......Page 78
Notes ......Page 81
Exercises ......Page 82
16.2 An extension due to Redheffer ......Page 84
16.3 Kober’s definition for the L^{\infty} case......Page 87
16.4.1 Connection with the Hilbert transform ......Page 89
16.4.2 Parseval-type formula for the Boas transform ......Page 91
16.4.4 Riesz-type bound for the Boas transform ......Page 92
16.4.5 Fourier transform of the Boas transform ......Page 93
16.4.6 Two theorems due to Boas ......Page 94
16.4.7 Inversion of the Boas transform ......Page 95
16.4.8 Generalization of the Boas transform ......Page 96
16.5 The bilinear Hilbert transform ......Page 98
16.7 The directional Hilbert transform ......Page 100
16.8 Hilbert transforms along curves ......Page 102
16.9 The ergodic Hilbert transform ......Page 103
16.10 The helical Hilbert transform ......Page 106
16.11 Some miscellaneous extensions of the Hilbert transform ......Page 107
Notes ......Page 109
Exercises ......Page 110
17.2 Linear systems ......Page 113
17.4 Stationary systems ......Page 119
17.5 Primitive statement of causality ......Page 120
17.6 The frequency domain ......Page 121
17.7 Connection to analyticity ......Page 123
17.7.1 A generalized response function ......Page 127
17.8 Application of a theorem due to Titchmarsh ......Page 130
17.9 An acausal example ......Page 133
17.10 The Paley-Wiener log-integral theorem ......Page 135
17.11 Extensions of the causality concept ......Page 142
17.12 Basic quantum scattering: causality conditions ......Page 145
17.13 Extension of Titchmarsh’s theorem for distributions ......Page 150
Notes ......Page 156
Exercises ......Page 157
18.1 Introductory ideas on signal processing ......Page 159
18.2 The Hilbert filter ......Page 161
18.3 The auto-convolution, cross-correlation, and auto-correlation functions ......Page 163
18.4 The analytic signal ......Page 166
18.5 Amplitude modulation ......Page 175
18.6 The frequency domain ......Page 178
18.7.1 The Heaviside function ......Page 179
18.7.2 The signum function ......Page 182
18.7.3 The rectangular pulse function ......Page 183
18.7.5 The sinc pulse function ......Page 185
18.8 The Hilbert transform of step functions and pulse forms ......Page 186
18.9 The fractional Hilbert transform: the Lohmann-Mendlovic-Zalevsky definition ......Page 187
18.10 The fractional Fourier transform ......Page 189
18.11 The fractional Hilbert transform: Zayed’s definition ......Page 199
18.12 The fractional Hilbert transform: the Cusmariu definition ......Page 200
18.13 The discrete fractional Fourier transform ......Page 203
18.14 The discrete fractional Hilbert transform ......Page 208
18.15 The fractional analytic signal ......Page 209
18.16 Empirical mode decomposition: the Hilbert-Huang transform ......Page 210
Notes ......Page 218
Exercises ......Page 220
19.1 Some background from classical electrodynamics ......Page 222
19.2 Kramers-Kronig relations: a simple derivation ......Page 224
19.3 Kramers-Kronig relations: a more rigorous derivation ......Page 230
19.4 An alternative approach to the Kramers-Kronig relations ......Page 237
19.5 Direct derivation of the Kramers-Kronig relations on the interval [0,8) ......Page 239
19.6 The refractive index: Kramers-Kronig relations ......Page 241
19.7 Application of Herglotz functions ......Page 248
19.8 Conducting materials ......Page 256
19.9 Asymptotic behavior of the dispersion relations ......Page 259
19.10 Sum rules for the dielectric constant ......Page 262
19.11 Sum rules for the refractive index ......Page 267
19.12 Application of some properties of the Hilbert transform ......Page 271
19.13 Sum rules involving weight functions ......Page 276
19.15 Light scattering: the forward scattering amplitude ......Page 279
Notes ......Page 285
Exercises ......Page 288
20.2 Dispersion relations for the normal-incident reflectance and phase ......Page 290
20.3 Sum rules for the reflectance and phase ......Page 301
20.4 The conductance: dispersion relations ......Page 305
20.5 The energy loss function: dispersion relations ......Page 307
20.6 The permeability: dispersion relations ......Page 309
20.7 The surface impedance: dispersion relations ......Page 312
20.8 Anisotropic media ......Page 316
20.9 Spatial dispersion ......Page 318
20.10 Fourier series representation ......Page 328
20.11 Fourier series approach to the reflectance ......Page 332
20.12 Fourier and allied integral representation ......Page 336
20.13 Integral inequalities ......Page 338
Notes ......Page 341
Exercises ......Page 342
21.1 Introduction ......Page 344
21.2 Circular polarization ......Page 345
21.3 The complex refractive indices N_+ and N_-......Page 347
21.4 Are there dispersion relations for the individual complex refractive indices N+ and N-? ......Page 354
21.5 Magnetic optical activity: Faraday effect and magnetic circular dichroism ......Page 357
21.6 Sum rules for magneto-optical activity ......Page 361
21.7 Magnetoreflectivity ......Page 363
21.8 Optical activity ......Page 368
21.9 Dispersion relations for optical activity ......Page 383
21.10 Sum rules for optical activity ......Page 384
Notes ......Page 386
Exercises ......Page 387
22.1 Introduction ......Page 389
22.2 Some types of nonlinear optical response ......Page 395
22.3 Classical description: the anharmonic oscillator ......Page 397
22.4 Density matrix treatment ......Page 400
22.5 Asymptotic behavior for the nonlinear susceptibility ......Page 410
22.6 One-variable dispersion relations for the nonlinear susceptibility ......Page 415
22.7 Experimental verification of the dispersion relations for the nonlinear susceptibility ......Page 422
22.8 Dispersion relations in two variables ......Page 424
22.9 n-dimensional dispersion relations ......Page 425
22.10 Situations where the dispersion relations do not hold ......Page 426
22.11 Sum rules for the nonlinear susceptibilities ......Page 430
22.13 The nonlinear refractive index and the nonlinear permittivity ......Page 433
Notes ......Page 440
Exercises ......Page 441
23.2.1 The Josephson junction ......Page 443
23.2.2 Absorption enhancement ......Page 447
23.3 The phase retrieval problem ......Page 448
23.4 X-ray crystallography ......Page 454
23.5.1 Potential scattering ......Page 459
23.5.2 Dispersion relations for potential scattering ......Page 462
23.5.3 Dispersion relations for electron-H atom scattering ......Page 465
23.6 Magnetic resonance applications ......Page 470
23.7 DISPA analysis ......Page 472
23.8 Electrical circuit analysis ......Page 474
23.9 Applications in acoustics ......Page 481
23.10 Viscoelastic behavior ......Page 484
23.11 Epilog ......Page 485
Notes ......Page 486
Exercises ......Page 488
Appendix 1 - Tables of selected Hilbert transforms......Page 490
Appendix 2 - Atlas of Hilbert transform pairs......Page 570
References......Page 583
Author index......Page 662
Subject index......Page 678
