本书主要包括傅立叶变换和分布理论、一些熟知函数的傅立叶变换、函数的卷积、拓扑矢量空间、局部凸空间、施瓦兹测试函数空间、分布的微积分学、黎曼-希尔伯特问题、柯西积分的推论、解决黎曼—希尔伯特问题的卡莱美方法、广义希尔伯特变换、n维希尔伯特变换、反演公式、广义n＋1维迪利克雷边界值问题、n维广义希尔伯特变换、希尔伯特问题等内容。
本书可以被研究生用作关于施瓦兹分布和周期分布的希尔伯特变换的教科书，也可以作为专著进行研究。
Preface
1 Some Background
1.1 Fourier Transforms and the Theory of Distributions
1.2 Fourier Transforms of L2 Functions
1.2.1 Fourier Transforms of Some Well-known Functions
1.3 Convolution of Functions
1.3.1 Differentiation of the Fourier Transform
1.4 Theory of Distributions
1.4.1 Topological Vector Spaces
1.4.2 Locally Convex Spaces
1.4.3 Schwartz Testing Function Space Its Topology and Distributions
1.4.4 The Calculus of Distribution
1.4.5 Distributional Differentiation
1.5 Primitive of Distributions
1.6 Characterization of Distributions of Compact Supports
1.7 Convolution of Distributions
1.8 The Direct Product of Distributions
1.9 The Convolution of Functions
1.10 Regularization of Distributions
1.11 The Continuity of the Convolution Process
1.12 Fourier Transforms and Tempered Distributions
1.12.1 The Testing Function Space S(Rn)
1.13 The Space of Distributions of Slow Growth SI(Rn)
1.14 A Boundedness Property of Distributions of Slow Growth and Its Structure Formula,
1.15 A Characterization Formula for Tempered Distributions
1.16 Fourier Transform of Tempered Distributions
1.17 Fourier Transform of Distributions in D'(Rn) Exercises
2 The Riemann-Hilbert Problem
2.1 Some Corollaries on Cauchy Integrals
2.2 Riemarm's Problem
2.2.1 The Hilbert Problem
2.2.2 Riemann-Hilbert problem
2.3 Carleman's Approach to Solving the Riemann-Hilbert Problem
2.4 The Hilbert Inversion Formula for Periodic Functions
2.5 The Hilbert Transform on the Real Line
2.6 Finite Hilbert Transform as Applied to Aerofoil Theories
2.7 The Riemann-Hilbert Problem Applied to Crack Problems
2.8 Reduction of a Griffith Crack Problem to the Hilbert problem
2.9 Further Applications of the Hilbert Transform
2.9.1 The Hiibert Transform
2.9.2 The Hibert Transform and the Dispersion Relations Exercises
3 The Hilbert Transform of Distributions in D Lp, 1
3.2 Classical Hilbert Transform
3.3 Schwartz Testing Function Space,D Lp, 1
3.4 The Hilbert Transform of Distributions in D Lp, 1
3.5 The Inversion Theorem
3.5.1 Some Examples and Applications
3.6 Approximate Hilbert Transform of Distributions
3.6.1 Analytic Representation
3.6.2 Distributional Representation of Analytic Functions
3.7 Existence and Uniqueness of the Solution to a Dirichlet Boundary-Value Problem
3.8 The Hilbert Problem for Distributions in D Lp, 1
3.8.2 The Hilbert Problem in D Lp, 1
4.1 Introduction
4.2 The Testing Function Space H(D) and Its Topology
4.3 Generalized Hilbert Transformation
4.4 An Intrinsic Definition of the Space H(D) and Its Topology
4.5 The Intrinsic Definition of the Space H(D)
4.5.1 The Intrinsic Definition of the Topology of H(D)
4.6 A Gel'fand-Shilov Technique for the Hilbert Transform
4.6.1 Gel'fand-Shilov Testing Function Space
4.6.2 The Topology of the Space (b
4.7 An Extension of the Gei'fand-Shilov Technique for the Hilbert Transform,
4.7.1 The Testing Function Space S
4.7.2 The Testing Function Space Z
4.7.3 The Hilbert Transform of Ultradistributions in Z
4.8 Distributional Hilbert Transforms in n-Dimensions
4.8.1 The Testing Function Space S1(R n)
4.8.2 The Testing Function Space Z1(R n)
4.8.3 The Testing Function Space Sn(R n)
4.8.4 The Testing Function Space Zn(R n)
4.8.5 The Strict Inductive Limit Topology of Zn(Rn) Exercises
5 n-Dimensional Hiibert Transform
5.1 Generalized n-Dimensional Hilbert Transform and Applications
5.1.1 Notation and Preliminaries
5.1.2 The Testing Function Space 23~(R n)
5.1.3 The Test Space X(R n)
5.2 The Hilbert Transform of a Test Function in X(R n)
5.2.1 The Hilbert Transform of Schwartz Distributions in D L(Ra),p > 1
5.3 Some Examples
5.4 Generalized (n + l)-Dimensional Dirichlet Boundary-Value Problems
5.5 The Hilbert Transform of Distributions in D L(Rn), p > 1 Its Inversion and Applications, 15
5.5.1 The n-Dimensional Hilbert Transform
5.5.2 Schwartz Testing Functions Space D(R n)
5.5.3 The Inversion Formula
5.5.4 The Topology on the Space Dtp(Rn)
5.5.5 The n-Dimensional Distributional Hilbert Transform
5.5.6 Calculus on ~D L(Rn)
5.5.7 The Testing Function Space HCD(R n))
5.5.8 The n-Dimensional Generalized Hilbert Transform
5.5.9 An Intrinsic Definition of the Space H(D(RN)) and Its Topology
Exercises
6 Further Applications of the Hilbert Transform the Hilbert ProblemuA Distributional Approach
6.1 Introduction
6.2 The Hilbert Problem
6.3 The Fourier Transform and the Hilbert Transform
6.4 Definitions and Preliminaries
6.5 The Action of the Fourier Transform on the Hilbert Transform and Vice Versa
6.6 Characterization of the Space F(So(R n))
6.7 The p-Norm of the Truncated Hilbert Transform
6.8 Operators on LP(R~) that Commute with Translations and Dilatations
6.9 Functions Whose Fourier Transforms Are Supported on Orthants
6.9.1 The Schwartz Distribution Space D L(R n)
6.9.2 An Approximate Hilbert Transform and Its Limit in LP(R n)
6.9.3 Complex Hilbert Transform
6.9.4 Distributional Representation of Holomorphic Functions
6.9.5 Action of the Fourier Transform on the Hilbert Transform
6.10 The Dirichlet Boundary-Value Problem, 2
6.11 Eigenvalues and Eigenfunctions of the Operator H Exercises
7 Periodic Distributions, Their Hilbert Transform and Applications
7.1 The Hilbert Transform of Periodic Distributions
7.1.1 Introduction
7.2 Definitions and Preliminaries
7.2.1 Testing Function Space P2r
7.2.2 The Space P~ of Periodic Distributions
7.3 Some Well-Known Operations on P
7.4 The Function Space L~ and Its Hilbert Transform, p > 1
7.5 The Inversion Formula
7.6 The Testing Function Space Q2r
7.7 The Hilbert Transform of Locally lntegrable and Periodic Function of Period
7.8 Approximate Hilbert Transform of Periodic Distributions
7.8.1 Introduction to Approximate Hilbert Transform
7.8.2 Notation and Preliminaries
7.9 A Structure Formula for Periodic Distributions
7.9.1 Applications
Exercises
Bibliography
Subject Index
Notation Index
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