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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