Generalized functions. Theory and technique.
2nd ed.

*(English)*Zbl 0893.46032
Boston, MA: Birkhäuser. ix, 462 p. (1998).

This is the second edition of the author’s book with the same title published in 1983 by Academic Press, Subsidiary of Harcourt Brace Jovanovich (Zbl 0538.46022). It continues to develop the same approach – to present the main ideas of the theory of Schwartz-Sobolev distributions on the main facts of it and using a lot of simple examples for illustrating these results. As the previous edition, the book is written in such a way that it can be used by physicists, theoretically inclined engineers and other scientists working with mathematical tools but not in mathematics. Besides it remains to be useful for students presenting a nice introduction into the area. One of the main achievements of the book is that it is allowed to become familiar with distributional approach without having a great knowledge in advanced analysis.

The book contains two parts. In first of them the theory of distributions is developed. The material is quite standard. The advantage of this text is in carefully gathered examples explaining how to use the corresponding properties. The theory presented in this part is only slightly revised with respect to the previous edition. But the series of examples and exercises is sufficiently enlarged.

The second part is devoted to applications. It contains more changes in comparison with the 83’ book. Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology. Some additional operators are discussed as well. A series of new calculations with generalized functions are involved, too. The “old” chapters 12, 13 are adjoint and presented now in the new chapter 12 “Applications to wave propagation”. Completely new is chapter 13 “Interplay between generalized functions and the theory of moments”, containing a distributional approach to asymptotic analysis and its applications. New applications of the theory of generalized functions are added in the last chapter 15. Among them are periodic distributions presented in the course of asymptotic analysis and applications of microlocal theory. All these changes certainly improve the text and make the book more useful for understanding the main approaches in the Schwartz-Sobolev theory and its numerous applications.

The book contains two parts. In first of them the theory of distributions is developed. The material is quite standard. The advantage of this text is in carefully gathered examples explaining how to use the corresponding properties. The theory presented in this part is only slightly revised with respect to the previous edition. But the series of examples and exercises is sufficiently enlarged.

The second part is devoted to applications. It contains more changes in comparison with the 83’ book. Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology. Some additional operators are discussed as well. A series of new calculations with generalized functions are involved, too. The “old” chapters 12, 13 are adjoint and presented now in the new chapter 12 “Applications to wave propagation”. Completely new is chapter 13 “Interplay between generalized functions and the theory of moments”, containing a distributional approach to asymptotic analysis and its applications. New applications of the theory of generalized functions are added in the last chapter 15. Among them are periodic distributions presented in the course of asymptotic analysis and applications of microlocal theory. All these changes certainly improve the text and make the book more useful for understanding the main approaches in the Schwartz-Sobolev theory and its numerous applications.

Reviewer: S.V.Rogozin (Minsk)

##### MSC:

46F10 | Operations with distributions and generalized functions |

46F12 | Integral transforms in distribution spaces |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

44A60 | Moment problems |

35Dxx | Generalized solutions to partial differential equations |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

34Axx | General theory for ordinary differential equations |

34Bxx | Boundary value problems for ordinary differential equations |