Generalized Functions Theory and Technique Second Edition

Transcription

1 Ram P. Kanwal Generalized Functions Theory and Technique Second Edition Birkhauser Boston Basel Berlin

2 Contents Preface to the Second Edition x Chapter 1. The Dirac Delta Function and Delta Sequences The heaviside function The Dirac delta function The delta sequences Aunitdipole The heaviside sequences 14 Exercises 15 Chapter 2. The Schwartz-Sobolev Theory of distributions Some introductory definitions Test functions Linear functionals and the Schwartz-Sobolev theory of distributions Examples Algebraic operations on distributions Analytic operations on distributions Examples The support and singular support of a distribution Exercises 45 Chapter 3. Additional Properties of Distributions Transformation properties of the delta distribution Convergence of distributions Delta sequences with parametric dependence Fourier series Examples The delta function as a Stieltjes integral 66 Exercises 67 Chapter 4. Distributions Defined by Divergent Integrals Introduction The pseudofunction #(*)/*",/i = 1,2, 3, Functions with algebraic singularity of order m Examples 81 Exercises 96

3 vi Contents Chapter 5. Distributional Derivatives of Functions with Jump 99 Discontinuities 5.1. Distributional derivatives in i?i Moving surfaces of discontinuity in /?, n > Surface distributions Various other representations First-order distributional derivatives Ill 5.6. Second-order distributional derivatives Higher-order distributional derivatives The two-dimensional case Examples The function Pf(l/r) and its derivatives 133 Chapter 6. Tempered Distributions and the Fourier Transform Preliminary concepts Distributions of slow growth (tempered distributions) The Fourier transform Examples 148 Exercises 168 Chapter 7. Direct Products and Convolutions of Distributions Definition of the direct product The direct product of tempered distributions The Fourier transform of the direct product of tempered distributions The convolution The role of convolution in the regularization of the distributions The dual spaces E and E' Examples The Fourier transform of a convolution Distributional solutions of integral equations 200 Exercises 205 Chapter 8. The Laplace Transform A brief discussion of the classical results The Laplace transform distributions The Laplace transform of the distributional derivatives and vice versa Examples 213 Exercises 218

4 Contents vii Chapter 9. Applications to Ordinary Differential Equations Ordinary differential operators Homogeneous differential equations Inhomogeneous differentational equations: the integral of a distribution Examples Fundamental solutions and Green's functions Second-order differential equations with constant coefficients Eigenvalue problems Second-order differential equations with variable coefficients Fourth-order differential equations Differential equations of nth order Ordinary differential equations with singular coefficients 246 Exercises 254 Chapter 10. Applications to Partial Differential Equations Introduction Classical and generalized solutions Fundamental solutions The Cauchy-Riemann operator The transport operator The Laplace operator The heat operator The Schrodinger operator The Helmholtz operator The wave operator The inhomogeneous wave equation The Klein-Gordon operator 286 Exercises 290 Chapter 11. Applications to Boundary Value Problems Poisson's equation Dumbbell-shaped bodies Uniform axial distributions Linear axial distributions Parabolic axial distributions, n = The fourth-order polynomial distribution, n = 7; spheroidal cavities 310

5 viii Contents The polarization tensor for a spheroid The virtual mass tensor for a spheroid The electric and magnetic polarizability tensors The distributional approach to scattering theory Stokes flow Displacement-type boundary value problems in elastostatistics The extension to elastodynamics Distributions on arbitrary lines Distributions on plane curves Distributions on a circular disk 342 Chapter 12. Applications to Wave Propagation Introduction The wave equation First-order hyperbolic systems Aerodynamic sound generation The Rankine-Hugoniot conditions Wave fronts that carry infinite singularities Kinematics of wavefronts Derivation of the transport theorems for wave fronts Propagation of wave fronts carrying multilayer densities Generalized functions with support on the light cone Examples 376 Chapter 13. Interplay Between Generalized Functions and the Theory of Moments The theory of moments Asymptotic approximation of integrals Applications to the singular perturbation theory Applications to number theory Distributional weight functions for orthogonal polynomials Convolution type integral equation revisited Further applications 403

6 Contents ix Chapter 14. Linear Systems Operators The step response The impulse response The response to an arbitrary input Generalized functions as impulse response functions The transfer function Discrete-time systems The sampling theorem 415 Chapter 15. Miscellaneous Topics Applications to probability and random processes Applications to economics Periodic distributions Applications to microlocal theory 446 References 449 Index 457