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Cambridge University Press 978-0-521-51720-1 - Hilbert Transforms, Valume 2 Frederick W. King Table of Contents More information

Contents

Preface List of symbols List of abbreviations

page xxi xxv xxxviii

Volume II 15 Hilbert transforms in E n 15.1 Definition of the Hilbert transform in E n 15.2 Definition of the n-dimensional Hilbert transform 15.3 The double Hilbert transform 15.4 Inversion property for the n-dimensional Hilbert transform 15.5 Derivative of the n-dimensional Hilbert transform 15.6 Fourier transform of the n-dimensional Hilbert transform 15.7 Relationship between the n-dimensional Hilbert transform and translation and dilation operators 15.8 The Parseval-type formula 15.9 Eigenvalues and eigenfunctions of the n-dimensional Hilbert transform 15.10 Periodic functions 15.11 A Calderón–Zygmund inequality 15.12 The Riesz transform 15.13 The n-dimensional Hilbert transform of distributions 15.14 Connection with analytic functions Notes Exercises

17 18 21 25 32 38 41 42

16 Some further extensions of the classical Hilbert transform 16.1 Introduction 16.2 An extension due to Redheffer 16.3 Kober’s definition for the L∞ case

44 44 44 47

1 1 5 8 10 11 12 14 16

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Contents 16.4

The Boas transform 16.4.1 Connection with the Hilbert transform 16.4.2 Parseval-type formula for the Boas transform 16.4.3 Iteration formula for the Boas transform 16.4.4 Riesz-type bound for the Boas transform 16.4.5 Fourier transform of the Boas transform 16.4.6 Two theorems due to Boas 16.4.7 Inversion of the Boas transform 16.4.8 Generalization of the Boas transform 16.5 The bilinear Hilbert transform 16.6 The vectorial Hilbert transform 16.7 The directional Hilbert transform 16.8 Hilbert transforms along curves 16.9 The ergodic Hilbert transform 16.10 The helical Hilbert transform 16.11 Some miscellaneous extensions of the Hilbert transform Notes Exercises 17 Linear systems and causality 17.1 Systems 17.2 Linear systems 17.3 Sequential systems 17.4 Stationary systems 17.5 Primitive statement of causality 17.6 The frequency domain 17.7 Connection to analyticity 17.7.1 A generalized response function 17.8 Application of a theorem due to Titchmarsh 17.9 An acausal example 17.10 The Paley–Wiener log-integral theorem 17.11 Extensions of the causality concept 17.12 Basic quantum scattering: causality conditions 17.13 Extension of Titchmarsh’s theorem for distributions Notes Exercises 18 The Hilbert transform of waveforms and signal processing 18.1 Introductory ideas on signal processing 18.2 The Hilbert filter 18.3 The auto-convolution, cross-correlation, and auto-correlation functions

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Contents 18.4 18.5 18.6 18.7

18.8 18.9

ix

The analytic signal Amplitude modulation The frequency domain Some useful step and pulse functions 18.7.1 The Heaviside function 18.7.2 The signum function 18.7.3 The rectangular pulse function 18.7.4 The triangular pulse function 18.7.5 The sinc pulse function The Hilbert transform of step functions and pulse forms The fractional Hilbert transform: the Lohmann–Mendlovic–Zalevsky definition The fractional Fourier transform The fractional Hilbert transform: Zayed’s definition The fractional Hilbert transform: the Cusmariu definition The discrete fractional Fourier transform The discrete fractional Hilbert transform The fractional analytic signal Empirical mode decomposition: the Hilbert–Huang transform

18.10 18.11 18.12 18.13 18.14 18.15 18.16 Notes Exercises

19 Kramers–Kronig relations 19.1 Some background from classical electrodynamics 19.2 Kramers–Kronig relations: a simple derivation 19.3 Kramers–Kronig relations: a more rigorous derivation 19.4 An alternative approach to the Kramers–Kronig relations 19.5 Direct derivation of the Kramers–Kronig relations on the interval [0, ∞) 19.6 The refractive index: Kramers–Kronig relations 19.7 Application of Herglotz functions 19.8 Conducting materials 19.9 Asymptotic behavior of the dispersion relations 19.10 Sum rules for the dielectric constant 19.11 Sum rules for the refractive index 19.12 Application of some properties of the Hilbert transform 19.13 Sum rules involving weight functions 19.14 Summary of sum rules for the dielectric constant and refractive index 19.15 Light scattering: the forward scattering amplitude Notes Exercises

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Contents 252 252

20 Dispersion relations for some linear optical properties 20.1 Introduction 20.2 Dispersion relations for the normal-incident reflectance and phase 20.3 Sum rules for the reflectance and phase 20.4 The conductance: dispersion relations 20.5 The energy loss function: dispersion relations 20.6 The permeability: dispersion relations 20.7 The surface impedance: dispersion relations 20.8 Anisotropic media 20.9 Spatial dispersion 20.10 Fourier series representation 20.11 Fourier series approach to the reflectance 20.12 Fourier and allied integral representation 20.13 Integral inequalities Notes Exercises

252 263 267 269 271 274 278 280 290 294 298 300 303 304

21 Dispersion relations for magneto-optical and natural optical activity 21.1 Introduction 21.2 Circular polarization 21.3 The complex refractive indices N+ and N− 21.4 Are there dispersion relations for the individual complex refractive indices N+ and N− ? 21.5 Magnetic optical activity: Faraday effect and magnetic circular dichroism 21.6 Sum rules for magneto-optical activity 21.7 Magnetoreflectivity 21.8 Optical activity 21.9 Dispersion relations for optical activity 21.10 Sum rules for optical activity Notes Exercises 22 Dispersion relations for nonlinear optical properties 22.1 Introduction 22.2 Some types of nonlinear optical response 22.3 Classical description: the anharmonic oscillator 22.4 Density matrix treatment 22.5 Asymptotic behavior for the nonlinear susceptibility 22.6 One-variable dispersion relations for the nonlinear susceptibility

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Contents 22.7

Experimental verification of the dispersion relations for the nonlinear susceptibility Dispersion relations in two variables n-dimensional dispersion relations Situations where the dispersion relations do not hold Sum rules for the nonlinear susceptibilities Summary of sum rules for the nonlinear susceptibilities The nonlinear refractive index and the nonlinear permittivity

22.8 22.9 22.10 22.11 22.12 22.13 Notes Exercises

xi 384 386 387 388 392 395 395 403 404

23 Some further applications of Hilbert transforms 23.1 Introduction 23.2 Hilbert transform spectroscopy 23.2.1 The Josephson junction 23.2.2 Absorption enhancement 23.3 The phase retrieval problem 23.4 X-ray crystallography 23.5 Electron–atom scattering 23.5.1 Potential scattering 23.5.2 Dispersion relations for potential scattering 23.5.3 Dispersion relations for electron–H atom scattering 23.6 Magnetic resonance applications 23.7 DISPA analysis 23.8 Electrical circuit analysis 23.9 Applications in acoustics 23.10 Viscoelastic behavior 23.11 Epilog Notes Exercises

406 406 406 406 410 411 417 422 422 425 428 433 435 437 444 447 448 449 451

Appendix 1 Table of selected Hilbert transforms

453

Appendix 2 Atlas of selected Hilbert transform pairs

534

References Author index Subject index

547 626 642

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Contents

Volume I 1

Introduction 1.1 Some common integral transforms 1.2 Definition of the Hilbert transform 1.3 The Hilbert transform as an operator 1.4 Diversity of applications of the Hilbert 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 Hölder 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 2.8 Some basic results from complex variable theory 2.8.1 Integration of analytic functions 2.8.2 Cauchy integral theorem 2.8.3 Cauchy integral formula 2.8.4 Jordan’s lemma 2.8.5 The Laurent expansion 2.8.6 The Cauchy residue theorem 2.8.7 Entire functions 2.9 Conformal mapping 2.10 Some functional analysis basics 2.10.1 Hilbert space 2.10.2 The Hardy space H p 2.10.3 Topological space 2.10.4 Compact operators 2.11 Lebesgue measure and integration 2.11.1 The notion of measure 2.12 Theorems due to Fubini and Tonelli 2.13 The Hardy–Poincaré–Bertrand formula

11 11 11 12 13 14 14 15 16 19 19 19 21 21 22 23 27 29 30 30 31 33 34 37 39 42 43 44 45 45 48 55 57

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Contents 2.14 2.15

2.16

Riemann–Lebesgue lemma Some elements of the theory of distributions 2.15.1 Generalized functions as sequences of functions 2.15.2 Schwartz distributions Summation of series: convergence accelerator techniques 2.16.1 Richardson extrapolation 2.16.2 The Levin sequence transformations

Notes Exercises

xiii 61 63 65 68 70 71 74 77 80

3

Derivation of the Hilbert transform relations 3.1 Hilbert transforms – basic forms 3.2 The Poisson integral for the half plane 3.3 The Poisson integral for the disc 3.3.1 The Poisson kernel for the disc 3.4 Hilbert transform on the real line 3.4.1 Conditions on the function f 3.4.2 The Phragmén–Lindelöf theorem 3.4.3 Some examples 3.5 Transformation to other limits 3.6 Cauchy integrals 3.7 The Plemelj formulas 3.8 Inversion formula for a Cauchy integral 3.9 Hilbert transform on the circle 3.10 Alternative approach to the Hilbert transform on the circle 3.11 Hardy’s approach 3.11.1 Hilbert transform on R 3.12 Fourier integral approach to the Hilbert transform on R 3.13 Fourier series approach 3.14 The Hilbert transform for periodic functions 3.15 Cancellation behavior for the Hilbert transform Notes Exercises

83 83 85 89 91 94 96 100 101 104 107 111 112 114 115 118 120 122 129 132 135 141 142

4

Some basic properties of the Hilbert transform 4.1 Introduction 4.1.1 Complex conjugation property 4.1.2 Linearity 4.2 Hilbert transforms of even or odd functions 4.3 Skew-symmetric character of Hilbert transform pairs 4.4 Inversion property

145 145 145 145 146 147 148

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Contents 4.5

4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

Scale changes 4.5.1 Linear scale changes 4.5.2 Some nonlinear scale transformations for the Hilbert transform Translation, dilation, and reflection operators The Hilbert transform of the product xn f (x) The Hilbert transform of derivatives Convolution property Titchmarsh formulas of the Parseval type Unitary property of H Orthogonality property Hilbert transforms via series expansion The Hilbert transform of a product of functions The Hilbert transform product theorem (Bedrosian’s theorem) A theorem due to Tricomi Eigenvalues and eigenfunctions of the Hilbert transform operator Projection operators A theorem due to Akhiezer The Riesz inequality The Hilbert transform of functions in L1 and in L∞ 4.21.1 The L∞ case Connection between Hilbert transforms and causal functions The Hardy–Poincaré–Bertrand formula revisited A theorem due to McLean and Elliott The Hilbert–Stieltjes transform A theorem due to Stein and Weiss

4.22 4.23 4.24 4.25 4.26 Notes Exercises 5

Relationship between the Hilbert transform and some common transforms 5.1 Introduction 5.2 Fourier transform of the Hilbert transform 5.3 Even and odd Hilbert transform operators 5.4 The commutator [F, H] 5.5 Hartley transform of the Hilbert transform 5.6 Relationship between the Hilbert transform and the Stieltjes transform 5.7 Relationship between the Laplace transform and the Hilbert transform 5.8 Mellin transform of the Hilbert transform 5.9 The Fourier allied integral

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Contents

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5.10 The Radon transform Notes Exercises

277 285 286

6 The Hilbert transform of periodic functions 6.1 Introduction 6.2 Approach using infinite product expansions 6.3 Fourier series approach 6.4 An operator approach to the Hilbert transform on the circle 6.5 Hilbert transforms of some standard kernels 6.6 The inversion formula 6.7 Even and odd periodic functions 6.8 Scale changes 6.9 Parseval-type formulas 6.10 Convolution property 6.11 Connection with Fourier transforms 6.12 Orthogonality property 6.13 Eigenvalues and eigenfunctions of the Hilbert transform operator 6.14 Projection operators 6.15 The Hardy–Poincaré–Bertrand formula 6.16 A theorem due to Privalov 6.17 The Marcel Riesz inequality 6.18 The partial sum of a Fourier series 6.19 Lusin’s conjecture Notes Exercises

288 288 289 291 292 296 301 303 304 305 307 309 310 311 312 313 316 318 323 325 328 329

7

331 331 335 335 340 343

Inequalities for the Hilbert transform 7.1 The Marcel Riesz inequality revisited 7.1.1 Hilbert’s integral 7.2 A Kolmogorov inequality 7.3 A Zygmund inequality 7.4 A Bernstein inequality 7.5 The Hilbert transform of a function having a bounded integral and derivative 7.6 Connections between the Hilbert transform on R and T 7.7 Weighted norm inequalities for the Hilbert transform 7.8 Weak-type inequalities 7.9 The Hardy–Littlewood maximal function 7.10 The maximal Hilbert transform function 7.11 A theorem due to Helson and Szegö 7.12 The Ap condition

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Contents 7.13 7.14 7.15

A theorem due to Hunt, Muckenhoupt, and Wheeden 7.13.1 Weighted norm inequalities for He and Ho Weighted norm inequalities for the Hilbert transform of functions with vanishing moments Weighted norm inequalities for the Hilbert transform with two weights Some miscellaneous inequalities for the Hilbert transform

7.16 Notes Exercises

395 399 402 403 408 413 416

8 Asymptotic behavior of the Hilbert transform 8.1 Asymptotic expansions 8.2 Asymptotic expansion of the Stieltjes transform 8.3 Asymptotic expansion of the one-sided Hilbert transform Notes Exercises

419 419 420 422 434 434

9

438 438 438

Hilbert transforms of some special functions 9.1 Hilbert transforms of special functions 9.2 Hilbert transforms involving Legendre polynomials 9.3 Hilbert transforms of the Hermite polynomials with a Gaussian weight 9.4 Hilbert transforms of the Laguerre polynomials with a weight function e−x 9.5 Other orthogonal polynomials 9.6 Bessel functions of the first kind 9.7 Bessel functions of the first and second kind for non-integer index 9.8 The Struve function 9.9 Spherical Bessel functions 9.10 Modified Bessel functions of the first and second kind 9.11 The cosine and sine integral functions 9.12 The Weber and Anger functions Notes Exercises

10 Hilbert transforms involving distributions 10.1 Some basic distributions 10.2 Some important spaces for distributions 10.3 Some key distributions 10.4 The Fourier transform of some key distributions 10.5 A Parseval-type formula approach to HT

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Contents 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14

Convolution operation for distributions Convolution and the Hilbert transform Analytic representation of distributions The inversion formula The derivative property The Fourier transform connection Periodic distributions: some preliminary notions The Hilbert transform of periodic distributions The Hilbert transform of ultradistributions and related ideas

Notes Exercises 11 The finite Hilbert 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 Hilbert transform for the interval (0, 1) 11.5 Inversion by a Fourier series approach 11.6 The Riemann problem 11.7 The Hilbert problem 11.8 The Riemann–Hilbert problem 11.8.1 The index of a function 11.9 Carleman’s approach 11.10 Some basic properties of the finite Hilbert transform 11.10.1 Even–odd character 11.10.2 Inversion property 11.10.3 Scale changes 11.10.4 Finite Hilbert transform of the product x n f (x) 11.10.5 Derivative of the finite Hilbert transform 11.10.6 Convolution property 11.10.7 Fourier transform of the finite Hilbert transform 11.10.8 Parseval-type identities 11.10.9 Orthogonality property 11.10.10 Eigenfunctions and eigenvalues of the finite Hilbert transform operator 11.11 Finite Hilbert transform of the Legendre polynomials 11.12 Finite Hilbert transform of the Chebyshev polynomials

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Contents

11.13 Contour integration approach to the derivation of some finite Hilbert transforms 11.14 The thin airfoil problem 11.15 The generalized airfoil problem 11.16 The cofinite Hilbert transform Notes Exercises

570 577 582 582 584 585

12 Some singular integral equations 12.1 Introduction 12.2 Fredholm equations of the first kind 12.3 Fredholm equations of the second kind 12.4 Fredholm equations of the third kind 12.5 Fourier transform approach to solving singular integral equations 12.6 A finite Hilbert transform integral equation 12.7 The one-sided Hilbert transform 12.7.1 Eigenfunctions and eigenvalues of the one-sided Hilbert transform operator 12.8 Fourier transform approach to the inversion of the one-sided Hilbert transform 12.9 An inhomogeneous singular integral equation for H1 12.10 A nonlinear singular integral equation 12.11 The Peierls–Nabarro equation 12.12 The sine–Hilbert equation 12.13 The Benjamin–Ono equation 12.13.1 Conservation laws 12.14 Singular integral equations involving distributions Notes Exercises

588 588 589 592 594 599 601 609

13 Discrete Hilbert transforms 13.1 Introduction 13.2 The discrete Fourier transform 13.3 Some properties of the discrete Fourier transform 13.4 Evaluation of the DFT 13.5 Relationship between the DFT and the Fourier transform 13.6 The Z transform 13.7 Z transform of a product 13.8 The Hilbert transform of a discrete time signal 13.9 Z transform of a causal sequence 13.10 Fourier transform of a causal sequence 13.11 The discrete Hilbert transform in analysis

637 637 637 640 641 643 644 647 649 652 656 660

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Contents

xix

13.12 Hilbert’s inequality 13.13 Alternative approach to the discrete Hilbert transform 13.14 Discrete analytic functions 13.15 Weighted discrete Hilbert transform inequalities Notes Exercises

661 666 675 679 680 681

14 Numerical evaluation of Hilbert transforms 14.1 Introduction 14.2 Some elementary transformations for Cauchy principal value integrals 14.3 Some classical formulas for numerical quadrature 14.3.1 A Maclaurin-type formula 14.3.2 The trapezoidal rule 14.3.3 Simpson’s rule 14.4 Gaussian quadrature: some basics 14.5 Gaussian quadrature: implementation procedures 14.6 Specialized Gaussian quadrature: application to the Hilbert transform 14.6.1 Error estimates 14.7 Specialized Gaussian quadrature: application to He and Ho 14.8 Numerical integration of the Fourier transform 14.9 The fast Fourier transform: numerical implementation 14.10 Hilbert transform via the fast Fourier transform 14.11 The Hilbert transform via the allied Fourier integral 14.12 The Hilbert transform via conjugate Fourier series 14.13 The Hilbert transform of oscillatory functions 14.14 An eigenfunction expansion 14.15 The finite Hilbert transform Notes Exercises Appendix 14.A Appendix 14.B

684 684

References Author index Subject index

745 824 840

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