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Hilbert transforms. Vol. 1 /
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 su...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge, UK ; New York :
Cambridge University Press,
2009.
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| Colección: | Encyclopedia of mathematics and its applications ;
volume 124. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; Dedication; Contents; Preface; List of symbols; List of abbreviations; 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; 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 inequality2.6 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 Hp; 2.10.3 Topological space
- 2.10.4 Compact operators2.11 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; 2.14 Riemann
- Lebesgue lemma; 2.15 Some elements of the theory of distributions; 2.15.1 Generalized functions as sequences of functions; 2.15.2 Schwartz distributions; 2.16 Summation of series: convergence accelerator techniques; 2.16.1 Richardson extrapolation; 2.16.2 The Levin sequence transformations; Notes; Exercises; 3 Derivation of the Hilbert transform relations; 3.1 Hilbert transforms
- basic forms
- 3.2 The Poisson integral for the half plane3.3 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 bold0mu mumu RRRawRRRR3.13 Fourier series approach; 3.14 The Hilbert transform for periodic functions; 3.15 Cancellation behavior for the Hilbert transform; Notes; Exercises; 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; 4.5 Scale changes; 4.5.1 Linear scale changes