1 Mathematical Preliminaries.- 1.1 Properties of Analytic Functions.- Problems for Section 1.1.- 1.2 Hilbert Transforms.- Problems for Section 1.

2.- 1.3 Riemann--Hilbert Problems.- 1.3.1 Some Definitions.- 1.3.

2 The Index of a Function.- 1.3.3 Homogeneous Problem.- 1.3.4 Solution of the Non-Homogeneous Problem.- Problems for Section 1.

3.- 1.4 Fourier Transforms.- Problems for Section 1.4.- 1.5 Laplace Transforms.- Problems for Section 1.

5.- 1.6 Fourier Series.- Problems for Section 1.6.- 1.7 Transformations of Functions.- Problems for Section 1.

7.- 1.8 Spaces of Analytic Functions.- 1.8.1 Functions Analytic on the Unit Disc.- 1.8.

2 Functions Analytic on ?+.- 1.8.3 Functions Analytic in the Strip Dd.- Problems for Section 1.8.- 1.9 The Paley--Wiener Theorem.

- Problems for Section 1.9.- 1.10 The Cardinal Function.- Problems for Section 1.10.- Historical Remarks on Chapter 1.- 2 Polynomial Approximation.

- 2.1 Chebyshev Polynomials.- Problems for Section 2.1.- 2.2 Discrete Fourier Polynomials.- Problems for Section 2.2.

- 2.3 The Lagrange Polynomial.- Problems for Section 2.3.- 2.4 Faber Polynomials.- Problems for Section 2.4.

- Historical Remarks on Chapter 2.- 3 Sinc Approximation in Strip.- 3.1 Sinc Approximation in Dd.- Problems for Section 3.1.- 3.2 Sinc Quadrature on (??, ?).

- Problems for Section 3.2.- 3.3 Discrete Fourier Transforms on (??,?).- Problems for Section 3.3.- 3.4 Cauchy-Like Transforms on (??,?).

- Problems for Section 3.4.- 3.5 Approximation of Derivatives in Dd.- Problems for Section 3.5.- 3.6 The Indefinite Integral on (??, ?).

- Problems for Section 3.6.- Historical Remarks on Chapter 3.- 4 Sinc Approximation on ?.- 4.1 Basic Definitions.- Problems for Section 4.1.

- 4.2 Interpolation and Quadrature on ?.- Problems for Section 4.2.- 4.3 Hilbert and Related Transforms on ?.- Problems for Section 4.3.

- 4.4 Approximation of Derivatives on ?.- Problems for Section 4.4.- 4.5 Indefinite Integral Over ?.- Problems for Section 4.5.

- 4.6 Indefinite Convolution Over ?.- 4.6.1 Approximation Procedure.- 4.6.2 Formula Derivation.

- 4.6.3 Convergence.- 4.6.4 Applications.- Problems for Section 4.6.

- Historical Remarks on Chapter 4.- 5 Sinc-Related Methods.- 5.1 Introduction.- 5.2 Variations of the Sinc Basis.- Problems for Section 5.2.

- 5.3 Elliptic Function Interpolants.- Problems for Section 5.3.- 5.4 Inversion of the Laplace Transform.- Problems for Section 5.4.

- 5.5 Sinc-Like Rational Approximation.- Problems for Section 5.5.- 5.6 Rationals and Extrapolation.- 5.6.

1 Padé Approximation.- 5.6.2 Continued Fractions.- 5.6.3 The Epsilon Algorithm and Aitken''s ?2 Process.- 5.

6.4 Chebyshev Acceleration.- 5.6.5 Thiele''s Algorithm.- Problems for Section 5.6.- 5.

7 Heaviside and Filter Rationals.- 5.7.1 Approximation of the Heaviside Function.- 5.7.2 Approximation of the Filter Function.- 5.

7.3 Approximation of the Delta Function.- Problems for Section 5.7.- 5.8 Positive Base Approximation.- Problems for Section 5.8.

- Historical Remarks on Chapter 5.- 6 Integral Equations.- 6.1 Introduction.- 6.2 Mathematical Preliminaries.- 6.2.

1 Use of Functional Analysis.- 6.2.2 Approximation, Convergence, and Error.- 6.2.3 Fredholm Alternative.- 6.

2.4 Perturbed Equations.- 6.2.5 Tikhonov Regularization.- Problems for Section 6.2.- 6.

3 Reduction to Algebraic Equations.- 6.3.1 Galerkin Method.- 6.3.2 Nyström''s Method.- 6.

3.3 The Generalized Inverse Procedure.- 6.3.4 Errors in the Numerical Solution.- Problems for Section 6.3.- 6.

4 Volterra Integral Equations.- 6.4.1 Linear Volterra Equations.- 6.4.2 Non-Linear Equations via Neumann Series.- 6.

4.3 Non-Linear Equations by Newton''s Method.- Problems for Section 6.4.- 6.5 Potential Theory Problems.- 6.5.

1 Problem Description.- 6.5.2 Spaces for Sinc Approximation.- 6.5.3 Sins Approximation.- 6.

5.4 Properties of the Integral Equation.- 6.5.5 Galerkin Approximation.- 6.5.6 Several Surface Patches--Domain Decomposition.

- 6.5.7 An Explicit Example.- 6.5.8 Kernel Singularities and Integration.- Problems for Section 6.5.

- 6.6 Reduced Wave Equation on a Half-Space.- 6.6.1 Problem Description.- 6.6.2 Spaces for Sinc Approximation.

- 6.6.3 Sinc Approximation.- 6.6.4 Properties of the Integral Equation.- 6.6.

5 Galerkin Approximation.- 6.6.6 Evaluation of Moment Integrals.- 6.6.7 Numerical Evaluation of Solution.- 6.

6.8 An Explicit Example.- Problems for Section 6.6.- 6.7 Cauchy Singular Integral Equations.- 6.7.

1 The Problem.- 6.7.2 The Method of Regularization.- 6.7.3 Properties of the Fredholm Equation.- 6.

7.4 Approximation via Nyström''s Method.- Problems for Section 6.7.- 6.8 Convolution-Type Equations.- 6.8.

1 The Problems and Theoretical Solutions.- 6.8.2 Approximate Solution.- 6.8.3 Explicit Examples.- Problems for Section 6.

8.- 6.9 The Laplace Transform and Its Inversion.- Problems for Section 6.9.- Historical Remarks on Chapter 6.- 7 Differential Equations.- 7.

1 ODE-IVP.- 7.1.1 Linear Initial Value Problems.- 7.1.2 Non-Linear Initial Value Problems.- Problems for Section 7.

1.- 7.2 ODE-BVP.- 7.2.1 Sinc-Galerkin and Collocation.- 7.2.

2 Integration by Parts.- 7.2.3 Collocation and Integration by Parts.- 7.2.4 Convergence of Sinc-Galerkin Methods.- 7.

2.5 Non-Linear Equations.- 7.2.6 Non-Homogeneous Boundary Conditions.- 7.2.7 Symmetric Sinc-Galerkin Method.

- 7.2.8 More Examples.- Problems for Section 7.2.- 7.3 Analytic Solutions of PDE.- 7.

3.1 Analyticity of Solutions in All Variables.- 7.3.2 Analyticity for Sinc Approximation.- 7.3.3 Singularities Due to Corners and Edges.

- Problems for Section 7.3.- 7.4 Elliptic Problems.- Problems for Section 7.4.- 7.5 Hyperbolic Problems.

- Problems for Section 7.5.- 7.6 Parabolic Problems.- Problems for Section 7.6.- Historical Remarks on Chapter 7.- References.