Numerical Methods Based on Sinc and Analytic Functions

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.- 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.- 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 ?.- Problemsfor 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.- Problems for Section 5.6.- 5.7 Heaviside and Filter Rationals.- 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.- Problems for Section 6.2.- 6.3 Reduction to Algebraic Equations.- Problems for Section 6.3.- 6.4 Volterra Integral Equations.- Problems for Section 6.4.- 6.5 Potential Theory Problems.- Problems for Section 6.5.- 6.6 Reduced Wave Equation on a Half-Space.- Problems for Section 6.6.- 6.7 Cauchy Singular Integral Equations.- Problems for Section 6.7.- 6.8 Convolution-Type Equations.- 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.- Problems for Section 7.1.- 7.2 ODE-BVP.- Problems for Section 7.2.- 7.3 Analytic Solutions of PDE.- 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.