Generalized Functions Theory and Technique

1. The Dirac Delta Function and Delta Sequences.- 1.1. The heaviside function.- 1.2. The Dirac delta function.- 1.3. The delta sequences.- 1.4. A unit dipole.- 1.5. The heaviside sequences.- Exercises.- 2. The Schwartz-Sobolev Theory of distributions.- 2.1. Some introductory definitions.- 2.2. Test functions.- 2.3. Linear functionals and the Schwartz-Sobolev theory of distributions.- 2.4. Examples.- 2.5. Algebraic operations on distributions.- 2.6. Analytic operations on distributions.- 2.7. Examples.- 2.8. The support and singular support of a distribution.- Exercises.- 3. Additional Properties of Distributions.- 3.1. Transformation properties of the delta distribution.- 3.2. Convergence of distributions.- 3.3. Delta sequences with parametric dependence.- 3.4. Fourier series.- 3.5. Examples.- 3.6. The delta function as a Stieltjes integral.- Exercises.- 4. Distributions Defined by Divergent Integrals.- 4.1. Introduction.- 4.2. The pseudofunction H(x)/xn, n = 1, 2, 3,....- 4.3. Functions with algebraic singularity of order m.- 4.4. Examples.- Exercises.- 5. Distributional Derivatives of Functions with Jump Discontinuities.- 5.1. Distributional derivatives in R1.- 5.2. Moving surfaces of discontinuity in Rn, n ? 2.- 5.3. Surface distributions.- 5.4. Various other representations.- 5.5. First-order distributional derivatives.- 5.6. Second-order distributional derivatives.- 5.7. Higher-order distributional derivatives.- 5.8. The two-dimensional case.- 5.9. Examples.- 5.10. The function Pf(1/r) and its derivatives.- 6. Tempered Distributions and the Fourier Transform.- 6.1. Preliminary concepts.- 6.2. Distributions of slow growth (tempered distributions).- 6.3. The Fourier transform.- 6.4. Examples.- Exercises.- 7. Direct Products and Convolutions of Distributions.- 7.1.Definition of the direct product.- 7.2. The direct product of tempered distributions.- 7.3. The Fourier transform of the direct product of tempered distributions.- 7.4. The convolution.- 7.5. The role of convolution in the regularization of the distributions.- 7.6. The dual spaces E and E?.- 7.7. Examples.- 7.8. The Fourier transform of a convolution.- 7.9. Distributional solutions of integral equations.- Exercises.- 8. The Laplace Transform.- 8.1. A brief discussion of the classical results.- 8.2. The Laplace transform distributions.- 8.3. The Laplace transform of the distributional derivatives and vice versa.- 8.4. Examples.- Exercises.- 9. Applications to Ordinary Differential Equations.- 9.1. Ordinary differential operators.- 9.2. Homogeneous differential equations.- 9.3. Inhomogeneous differentational equations: the integral of a distribution.- 9.4. Examples.- 9.5. Fundamental solutions and Green’s functions.- 9.6. Second-order differential equations with constant coefficients.- 9.7. Eigenvalue problems.- 9.8. Second-order differential equations with variable coefficients.- 9.9. Fourth-order differential equations.- 9.10. Differential equations of nth order.- 9.11. Ordinary differential equations with singular coefficients.- Exercises.- 10. Applications to Partial Differential Equations.- 10.1. Introduction.- 10.2. Classical and generalized solutions.- 10.3. Fundamental solutions.- 10.4. The Cauchy-Riemann operator.- 10.5. The transport operator.- 10.6. The Laplace operator.- 10.7. The heat operator.- 10.8. The Schrödinger operator.- 10.9. The Helmholtz operator.- 10.10. The wave operator.- 10.11. The inhomogeneous wave equation.- 10.12. The Klein-Gordon operator.- Exercises.- 11. Applications to Boundary Value Problems.- 11.1. Poisson’s equation.- 11.2.Dumbbell-shaped bodies.- 11.3. Uniform axial distributions.- 11.4. Linear axial distributions.- 11.5. Parabolic axial distributions, n = 5.- 11.6. The fourth-order polynomial distribution, n = 7; spheroidal cavities.- 11.7. The polarization tensor for a spheroid.- 11.8. The virtual mass tensor for a spheroid.- 11.9. The electric and magnetic polarizability tensors.- 11.10. The distributional approach to scattering theory.- 11.11. Stokes flow.- 11.12. Displacement-type boundary value problems in elastostatistics.- 11.13. The extension to elastodynamics.- 11.14. Distributions on arbitrary lines.- 11.15. Distributions on plane curves.- 11.16. Distributions on a circular disk.- 12. Applications to Wave Propagation.- 12.1. Introduction.- 12.2. The wave equation.- 12.3. First-order hyperbolic systems.- 12.4. Aerodynamic sound generation.- 12.5. The Rankine-Hugoniot conditions.- 12.6. Wave fronts that carry infinite singularities.- 12.7. Kinematics of wavefronts.- 12.8. Derivation of the transport theorems for wave fronts.- 12.9. Propagation of wave fronts carrying multilayer densities.- 12.10. Generalized functions with support on the light cone.- 12.11. Examples.- 13. Interplay Between Generalized Functions and the Theory of Moments.- 13.1. The theory of moments.- 13.2. Asymptotic approximation of integrals.- 13.3. Applications to the singular perturbation theory.- 13.4. Applications to number theory.- 13.5. Distributional weight functions for orthogonal polynomials.- 13.6. Convolution type integral equation revisited.- 13.7. Further applications.- 14. Linear Systems.- 14.1. Operators.- 14.2. The step response.- 14.3. The impulse response.- 14.4. The response to an arbitrary input.- 14.5. Generalized functions as impulse response functions.- 14.6. The transfer function.- 14.7. Discrete-time systems.- 14.8. The sampling theorem.- 15. Miscellaneous Topics.- 15.1. Applications to probability and random processes.- 15.2. Applications to economics.- 15.3. Periodic distributions.- 15.4. Applications to microlocal theory.- References.