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

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## Ram Prakash Kanwal:

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

Published $\text {1998}$, **BirkhĂ¤user**

- 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 SchrĂ¶dinger 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