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Preface | p. xv |
One Dimensional Sinc Theory | p. 1 |
Introduction and Summary | p. 1 |
Some Introductory Remarks | p. 2 |
Uses and Misuses of Sinc | p. 5 |
Sampling over the Real Line | p. 7 |
Problems for Section 1.2 | p. 11 |
More General Sinc Approximation on R | p. 18 |
Infinite Term Sinc Approximation on R | p. 18 |
Finite Term Sinc Approximation on R | p. 25 |
Problems for Section 1.3 | p. 31 |
Sinc, Wavelets, Trigonometric and Algebraic Polynomials and Quadratures | p. 32 |
A General Theorem | p. 35 |
Explicit Special Cases on [0, 2¿] | p. 36 |
Wavelets and Trigonometric Polynomials | p. 40 |
Error of Approximation | p. 42 |
Algebraic Interpolation and Quadrature | p. 48 |
Wavelet Differentiation | p. 60 |
Wavelet Indefinite Integration | p. 62 |
Hilbert Transforms | p. 63 |
Discrete Fourier Transform | p. 65 |
Problems for Section 1.4 | p. 68 |
Sinc Methods on ¿ | p. 70 |
Sinc Approximation on a Finite Interval | p. 70 |
Sinc Spaces for Intervals and Arcs | p. 72 |
Important Explicit Transformations | p. 79 |
Interpolation on ¿ | p. 82 |
Sinc Approximation of Derivatives | p. 87 |
Sinc Collocation | p. 89 |
Sinc Quadrature | p. 90 |
Sinc Indefinite Integration | p. 92 |
Sinc Indefinite Convolution | p. 93 |
Laplace Transform Inversion | p. 100 |
More General 1 - d Convolutions | p. 101 |
Hilbert and Cauchy Transforms | p. 105 |
Analytic Continuation | p. 113 |
Initial Value Problems | p. 116 |
Wiener-Hopf Equations | p. 118 |
Problems for Section 1.5 | p. 120 |
Rational Approximation at Sinc Points | p. 125 |
Rational Approximation in M_{¿,ß,d(¿)} | p. 126 |
Thiele-Like Algorithms | p. 127 |
Problems for Section 1.6 | p. 128 |
Polynomial Methods at Sinc Points | p. 129 |
Sinc Polynomial Approximation on (0, 1) | p. 130 |
Polynomial Approximation on ¿ | p. 133 |
Approximation of the Derivative on ¿ | p. 134 |
Problems for Section 1.7 | p. 137 |
Sinc Convolution-BIE Methods for PDE & IE | p. 139 |
Introduction and Summary | p. 139 |
Some Properties of Green's Functions | p. 141 |
Directional Derivatives | p. 141 |
Integrals along Arcs | p. 142 |
Surface Integrals | p. 142 |
Some Green's Identities | p. 143 |
Problems for Section 2.2 | p. 150 |
Free-Space Green's Functions for PDE | p. 150 |
Heat Problems | p. 151 |
Wave Problems | p. 151 |
Helmholtz Equations | p. 152 |
Biharmonic Green's Functions | p. 154 |
Laplace Transforms of Green's Functions | p. 155 |
Transforms for Poisson Problems | p. 158 |
Transforms for Helmholtz Equations | p. 163 |
Transforms for Hyperbolic Problems | p. 168 |
Wave Equation in R^{3} × (0, T) | p. 169 |
Transforms for Parabolic Problems | p. 172 |
Navier-Stokes Equations | p. 173 |
Transforms for Biharmonic Green's Functions | p. 180 |
Problems for Section 2.4 | p. 184 |
Multi-Dimensional Convolution Based on Sinc | p. 187 |
Rectangular Region in 2 - d | p. 187 |
Rectangular Region in 3 - d | p. 191 |
Curvilinear Region in 2 - d | p. 192 |
Curvilinear Region in 3 - d | p. 199 |
Boundary Integral Convolutions | p. 207 |
Problems for Section 2.5 | p. 209 |
Theory of Separation of Variables | p. 209 |
Regions and Function Spaces | p. 210 |
Analyticity and Separation of Variables | p. 222 |
Problems for Section 2.6 | p. 242 |
Explicit 1-d Programs Solutions via Sinc-Pack | p. 243 |
Introduction and Summary | p. 243 |
Sinc Interpolation | p. 245 |
Sinc Points Programs | p. 246 |
Sinc Basis Programs | p. 248 |
Interpolation and Approximation | p. 251 |
Singular, Unbounded Functions | p. 257 |
Problems for Section 3.2 | p. 257 |
Approximation of Derivatives | p. 258 |
Problems for Section 3.3 | p. 261 |
Sinc Quadrature | p. 262 |
Problems for Section 3.4 | p. 265 |
Sinc Indefinite Integration | p. 266 |
Problems for Section 3.5 | p. 268 |
Sinc Indefinite Convolution | p. 270 |
Problems for Section 3.6 | p. 274 |
Laplace Transform Inversion | p. 275 |
Problems for Section 3.7 | p. 278 |
Hilbert and Cauchy Transforms | p. 280 |
Problems for Section 3.8 | p. 283 |
Sinc Solution of ODE | p. 284 |
Nonlinear ODE-IVP on (0, T) via Picard | p. 285 |
Linear ODE-IVP on (0, T) via Picard | p. 286 |
Linear ODE-IVP on (0, T) via Direct Solution | p. 289 |
Second-Order Equations | p. 292 |
Wiener-Hopf Equations | p. 298 |
Wavelet Examples | p. 300 |
Wavelet Approximations | p. 302 |
Wavelet Sol'n of a Nonlinear ODE via Picard | p. 309 |
Explicit Program Solutions of PDE via Sinc-Pack | p. 315 |
Introduction and Summary | p. 315 |
Elliptic PDE | p. 315 |
Harmonic Sinc Approximation | p. 315 |
A Poisson-Dirichlet Problem over R^{2} | p. 320 |
A Poisson-Dirichlet Problem over a Square | p. 323 |
Neumann to a Dirichlet Problem on Lemniscate | p. 332 |
A Poisson Problem over a Curvilinear Region in R^{2} | p. 338 |
A Poisson Problem over R^{3} | p. 350 |
Hyperbolic PDE | p. 356 |
Solving a Wave Equation Over R^{3} × (0, T) | p. 356 |
Solving Helmholtz Equation | p. 364 |
Parabolic PDE | p. 365 |
A Nonlinear Population Density Problem | p. 365 |
Navier-Stokes Example | p. 383 |
Performance Comparisons | p. 404 |
The Problems | p. 404 |
The Comparisons | p. 405 |
Directory of Programs | p. 409 |
Wavelet Formulas | p. 409 |
One Dimensional Sinc Programs | p. 410 |
Standard Sinc Transformations | p. 410 |
Sinc Points and Weights | p. 411 |
Interpolation at Sinc Points | p. 412 |
Derivative Matrices | p. 413 |
Quadrature | p. 414 |
Indefinite Integration | p. 416 |
Indefinite Convolution | p. 416 |
Laplace Transform Inversion | p. 418 |
Hilbert Transform Programs | p. 418 |
Cauchy Transform Programs | p. 419 |
Analytic Continuation | p. 420 |
Cauchy Transforms | p. 421 |
Initial Value Problems | p. 421 |
Wiener-Hopf Equations | p. 422 |
Multi-Dimensional Laplace Transform Programs | p. 422 |
Q - Function Program | p. 422 |
Transf. for Poisson Green's Functions | p. 422 |
Transforms of Helmholtz Green's Function | p. 423 |
Transforms of Hyperbolic Green's Functions | p. 423 |
Transforms of Parabolic Green's Functions | p. 424 |
Transf. of Navier-Stokes Green's Functions | p. 425 |
Transforms of Biharmonic Green's Functions | p. 425 |
Example Programs for PDE Solutions | p. 426 |
Bibliography | p. 429 |
Index | p. 461 |
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