# Book:Ram Prakash Kanwal/Generalized Functions: Theory and Technique/Second Edition

## Ram Prakash Kanwal: Generalized Functions: Theory and Technique (2nd Edition)

Published $\text {1998}$, Birkhäuser

ISBN 1-4684-0035-9

### Contents

Preface to the second edition
preface to the first edition

Chapter 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

Chapter 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

Chapter 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

Chapter 4. Distributions Defined by Divergent Integrals
4.1. Introduction
4.2. The pseudofunction $\map H x /x^n , n = 1,2,3,\ldots$
4.3. Functions with algebraic singularity of order $m$
4.4. Examples
Exercises

Chapter 5. Distributional Derivatives of Functions with Jump Discontinuities

5.1. Distributional derivatives in $R_1$
5.2. Moving surfaces of discontinuity in $R_n, n \ge 2$
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 $\map {Pf} {1/r}$ and its derivatives

Chapter 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

Chapter 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

Chapter 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

Chapter 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 $n$th order
9.11. Ordinary differential equations with singular coefficients
Exercises

Chapter 10. Applications to Partial Differential Equations
10.1. Introduction
10.2. Classical and generalized solutions
10.3. Fundamental solutions
lO.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

Chapter 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 tensorfor 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

Chapter 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 wave fronts
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

Chapter 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

Chapter 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

Chapter 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
Index