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…mehr
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.- 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.
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.
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