The Hilbert transform of Schwartz distributions and applications:
This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Wiley
1996
|
| Schriftenreihe: | Pure and applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for mathematicians, applied scientists, and engineers. The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others Innovative at every step, J.N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions. The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of R[superscript n] Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology. The book incudes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science |
| Beschreibung: | XVI, 262 S. |
| ISBN: | 0471033731 |
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| 520 | 3 | |a This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for mathematicians, applied scientists, and engineers. The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others | |
| 520 | |a Innovative at every step, J.N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions. The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of R[superscript n] | ||
| 520 | |a Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology. The book incudes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science | ||
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Datensatz im Suchindex
| _version_ | 1804125338056261632 |
|---|---|
| adam_text | CONTENTS
Preface xiii
1. Some Background 1
1.1. Fourier Transforms and the Theory of Distributions, 1
1.2. Fourier Transforms of L2 Functions, 4
1.2.1. Fourier Transforms of Some Well known Functions, 4
1.3. Convolution of Functions, 7
1.3.1. Differentiation of the Fourier Transform, 12
1.4. Theory of Distributions, 12
1.4.1. Topological Vector Spaces, 13
1.4.2. Locally Convex Spaces, 21
1.4.3. Schwartz Testing Function Space:
Its Topology and Distributions, 23
1.4.4. The Calculus of Distribution, 29
1.4.5. Distributional Differentiation, 31
1.5. Primitive of Distributions, 31
1.6. Characterization of Distributions of Compact Supports, 32
1.7. Convolution of Distributions, 33
1.8. The Direct Product of Distributions, 34
1.9. The Convolution of Functions, 36
1.10. Regularization of Distributions, 39
1.11. The Continuity of the Convolution Process, 39
1.12. Fourier Transforms and Tempered Distributions, 40
1.12.1. The Testing Function Space S(R ), 40
1.13. The Space of Distributions of Slow Growth S W), 41
7li
viii CONTENTS
1.14. A Boundedness Property of Distributions of Slow Growth
and Its Structure Formula, 41
1.15. A Characterization Formula for Tempered Distributions, 42
1.16. Fourier Transform of Tempered Distributions, 44
1.17. Fourier Transform of Distributions in D (IR ), 49
Exercises, 51
2. The Riemann Hilbert Problem 54
2.1. Some Corollaries on Cauchy Integrals, 54
2.2. Riemann s Problem, 56
2.2.1. The Hilbert Problem, 58
2.2.2. Riemann Hilbert Problem, 58
2.3. Carleman s Approach to Solving the Riemann Hilbert
Problem, 58
2.4. The Hilbert Inversion Formula for Periodic Functions, 66
2.5. The Hilbert Transform on the Real Line, 75
2.6. Finite Hilbert Transform as Applied to Aerofoil Theories, 82
2.7. The Riemann Hilbert Problem Applied to Crack Problems, 84
2.8. Reduction of a Griffith Crack Problem to the Hilbert Problem, 85
2.9. Further Applications of the Hilbert Transform, 86
2.9.1. The Hilbert Transform, 86
2.9.2. The Hibert Transform and the Dispersion Relations, 86
Exercises, 87
3. The Hilbert Transform of Distributions in V^, 1 p oo 89
3.1. Introduction, 89
3.2. Classical Hilbert Transform, 91
3.3. Schwartz Testing Function Space, Dlp, Kp °o, 93
3.3.1. The Topology on the Space T)t, 93
3.4. The Hilbert Transform of Distributions in T ^p, Kp °°, 96
3.4.1. Regular Distribution in ©^, 96
3.5. The Inversion Theorem, 97
3.5.1. Some Examples and Applications, 98
3.6. Approximate Hilbert Transform of Distributions, 100
3.6.1. Analytic Representation, 103
3.6.2. Distributional Representation of Analytic Functions, 104
3.7. Existence and Uniqueness of the Solution to a
Dirichlet Boundary Value Problem, 107
CONTENTS ix
3.8. The Hilbert Problem for Distributions in £ £,, 110
3.8.1. Description of the Problem, 110
3.8.2. The Hilbert Problem in T t, Kp » ill
Exercises, 113
4. The Hilbert Transform of Schwartz Distributions 114
4.1. Introduction, 114
4.2. The Testing Function Space H( T ) and Its Topology, 115
4.3. Generalized Hilbert Transformation, 116
4.4. An Intrinsic Definition of the Space H(D) and Its Topology, 118
4.5. The Intrinsic Definition of the Space H(T ), 121
4.5.1. The Intrinsic Definition of the Topology of H(£ ), 121
4.6. A Gel fand Shilov Technique for the Hilbert Transform, 121
4.6.1. Gel fand Shilov Testing Function Spaced, 122
4.6.2. The Topology of the Spaced, 124
4.7. An Extension of the Gel fand Shilov Technique
for the Hilbert Transform, 125
4.7.1. The Testing Function Space Si, 126
4.7.2. The Testing Function Space Zi, 126
4.7.3. The Hilbert Transform of Ultradistributions in Z{, 127
4.8. Distributional Hilbert Transforms in n Dimensions, 131
4.8.1. The Testing Function Space 5i(R ), 131
4.8.2. The Testing Function Space Z^R ), 131
4.8.3. The Testing Function Space SV(U ), 132
4.8.4. The Testing Function Space Z,(RB), 132
4.8.5. The Strict Inductive Limit Topology of ZV(W), 132
Exercises, 136
5. it Dimensional Hilbert Transform 138
5.1. Generalized M Dimensional Hilbert Transform and Applications, 138
5.1.1. Notation and Preliminaries, 138
5.1.2. The Testing Function Space D^R ), 138
5.1.3. The Test Space X(Rn), 139
5.2. The Hilbert Transform of a Test Function in X(Un), 142
5.2.1. The Hilbert Transform of Schwartz Distributions in
V]p(W),p 1, 145
5.3. Some Examples, 147
5.4. Generalized (n + 1) Dimensional Dirichlet
Boundary Value Problems, 149
x CONTENTS
5.5. The Hilbert Transform of Distributions in V{j,(W), p 1,
Its Inversion and Applications, 151
5.5.1. The n Dimensional Hilbert Transform, 152
5.5.2. Schwartz Testing Functions Space T (U ), 152
5.5.3. The Inversion Formula, 153
5.5.4. The Topology on the Space T u(W), 154
5.5.5. The n Dimensional Distributional Hilbert Transform, 156
5.5.6. Calculus on D^R ), 157
5.5.7. The Testing Function Space H(V(Un)), 159
5.5.8. The n Dimensional Generalized Hilbert Transform, 160
5.5.9. An Intrinsic Definition of the Space H(T (Un))
and Its Topology, 161
Exercises, 168
6. Further Applications of the Hilbert Transform,
the Hilbert Problem—A Distributional Approach 170
6.1. Introduction, 170
6.2. The Hilbert Problem, 174
6.3. The Fourier Transform and the Hilbert Transform, 178
6.4. Definitions and Preliminaries, 180
6.5. The Action of the Fourier Transform on the Hilbert Transform,
and Vice Versa, 181
6.6. Characterization of the Space F(S0(R )), 182
6.7. The p Norm of the Truncated Hilbert Transform, 183
6.8. Operators on LP(R ) that Commute with Translations
and Dilatations, 186
6.9. Functions Whose Fourier Transforms Are Supported on Orthants, 192
6.9.1. The Schwartz Distribution Space 7){j,(W), 194
6.9.2. An Approximate Hilbert Transform and Its Limit in I/(R ), 197
6.9.3. Complex Hilbert Transform, 202
6.9.4. Distributional Representation of Holomorphic Functions, 205
6.9.5. Action of the Fourier Transform on the Hilbert Transform, 206
6.10. The Dirichlet Boundary Value Problem, 213
6.11. Eigenvalues and Eigenfunctions of the Operator H, 214
Exercises, 215
7. Periodic Distributions, Their Hilbert Transform and Applications 217
7.1. The Hilbert Transform of Periodic Distributions, 217
7.1.1. Introduction, 217
CONTENTS xi
7.2. Definitions and Preliminaries, 222
7.2.1. Testing Function Space P2t, 222
7.2.2. The Space P{7 of Periodic Distributions, 223
7.3. Some Weil Known Operations on P±T, 228
7.4. The Function Space L£T and Its Hilbert Transform, p l, 228
7.5. The Inversion Formula, 230
7.6. The Testing Function Space Q2t, 231
7.7. The Hilbert Transform of Locally Integrable and Periodic Function of
Period 2t, 233
7.8. Approximate Hilbert Transform of Periodic Distributions, 236
7.8.1. Introduction to Approximate Hilbert Transform, 237
7.8.2. Notation and Preliminaries, 238
7.9. A Structure Formula for Periodic Distributions, 241
7.9.1. Applications, 245
Exercises, 247
Bibliography 249
Subject Index 255
Notation Index 259
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| spelling | Pandey, J. N. Verfasser aut The Hilbert transform of Schwartz distributions and applications J. N. Pandey New York [u.a.] Wiley 1996 XVI, 262 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics This book provides a modern and up-to-date treatment of the Hilbert transform of distributions and the space of periodic distributions. Taking a simple and effective approach to a complex subject, this volume is a first-rate textbook at the graduate level as well as an extremely useful reference for mathematicians, applied scientists, and engineers. The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others Innovative at every step, J.N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions. The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of R[superscript n] Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology. The book incudes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science Hilbert transform Schwartz distributions Hilbert-Transformation (DE-588)4375311-5 gnd rswk-swf Hilbert-Transformation (DE-588)4375311-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007253610&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pandey, J. N. The Hilbert transform of Schwartz distributions and applications Hilbert transform Schwartz distributions Hilbert-Transformation (DE-588)4375311-5 gnd |
| subject_GND | (DE-588)4375311-5 |
| title | The Hilbert transform of Schwartz distributions and applications |
| title_auth | The Hilbert transform of Schwartz distributions and applications |
| title_exact_search | The Hilbert transform of Schwartz distributions and applications |
| title_full | The Hilbert transform of Schwartz distributions and applications J. N. Pandey |
| title_fullStr | The Hilbert transform of Schwartz distributions and applications J. N. Pandey |
| title_full_unstemmed | The Hilbert transform of Schwartz distributions and applications J. N. Pandey |
| title_short | The Hilbert transform of Schwartz distributions and applications |
| title_sort | the hilbert transform of schwartz distributions and applications |
| topic | Hilbert transform Schwartz distributions Hilbert-Transformation (DE-588)4375311-5 gnd |
| topic_facet | Hilbert transform Schwartz distributions Hilbert-Transformation |
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