Numerical Methods Based on Sinc and Analytic Functions

Many mathematicians, scientists, and engineers are familiar with the Fast Fourier Transform, a method based upon the Discrete Fourier Transform. Perhaps not so many mathematicians, scientists, and engineers recognize that the Discrete Fourier Transform is one of a family of symbolic formulae called Sinc methods. Sinc methods are based upon the Sinc function, a wavelet-like function replete with identities which yield approximations to all classes of computational problems. Such problems include problems over finite, semi-infinite, or infinite domains, problems with singularities, and boundary layer problems. Written by the principle authority on the subject, this book introduces Sinc methods to the world of computation. It serves as an excellent research sourcebook as well as a textbook which uses analytic functions to derive Sinc methods for the advanced numerical analysis and applied approximation theory classrooms. Problem sections and historical notes are included.

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.