UID:
almafu_9959234318202883
Umfang:
1 online resource (xxxviii, 858 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88714-9
,
1-107-10169-7
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1-107-10412-2
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0-511-72145-5
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1-107-09608-1
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1-107-08979-4
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1-107-09302-3
Serie:
Encyclopedia of mathematics and its applications ; v. 124
Inhalt:
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.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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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
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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
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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
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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
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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
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4.5.2 Some nonlinear scale transformations for the Hilbert transform
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English
Weitere Ausg.:
ISBN 0-521-88762-3
Weitere Ausg.:
ISBN 1-306-14870-7
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9780511721458
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