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# Generalized Functions and Partial Differential Equations

## Paperback

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## Overview

This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research.

Geared toward upper-level undergraduates and graduate students, the text assumes a sound knowledge of both real and complex variables. Familiarity with the basic theory of functional analysis, especially normed spaces, is helpful but not necessary. An introductory chapter features helpful background on topological spaces. Applications to partial differential equations include a treatment of the Cauchy problem, the Goursat problem, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. Supplementary materials include end-of-chapter problems, bibliographical remarks, and a bibliography.

## Product Details

ISBN-13: | 9780486446103 |
---|---|

Publisher: | Dover Publications |

Publication date: | 12/10/2005 |

Series: | Dover Books on Mathematics Series |

Pages: | 352 |

Sales rank: | 1,288,296 |

Product dimensions: | 5.30(w) x 8.40(h) x 0.80(d) |

## Read an Excerpt

This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research.

Geared toward upper-level undergraduates and graduate students, the text assumes a sound knowledge of both real and complex variables. Familiarity with the basic theory of functional analysis, especially normed spaces, is helpful but not necessary. An introductory chapter features helpful background on topological spaces. Applications to partial differential equations include a treatment of the Cauchy problem, the Goursat problem, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. Supplementary materials include end-of-chapter problems, bibliographical remarks, and a bibliography.

## First Chapter

This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to solving a variety of problems in partial differential equations. A major portion of the text is based on material included in the books of L. Schwartz, who developed the theory of distributions, and in the books of Gelfand and Shilov, who deal with generalized functions of any class and their use in solving the Cauchy problem. In addition, the author provides applications developed through his own research.

Geared toward upper-level undergraduates and graduate students, the text assumes a sound knowledge of both real and complex variables. Familiarity with the basic theory of functional analysis, especially normed spaces, is helpful but not necessary. An introductory chapter features helpful background on topological spaces. Applications to partial differential equations include a treatment of the Cauchy problem, the Goursat problem, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. Supplementary materials include end-of-chapter problems, bibliographical remarks, and a bibliography.

## Table of Contents

1. Topological spaces and metric spaces

2. Linear topological spaces

3. Countably normed spaces

4. Continuous linear functionals

5. Weak and strong topologies

6. Perfect spaces

7. Linear operators

8. Inductive limits and unions of topological spaces

Problems

2 Spaces of Generalized Functions

1. Fundamental spaces and generalized functions

2. The spaces K{Mp}

3. The spaces Z{Mp}

4. Multiplication and the derivatives of generalized functions

5. Structure of generalized functions on K{Mp}

Problems

3 Theory of Distributions

1. Spaces of functions

2. Partition of unity

3. Definition and some properties of distributions

4. Derivatives of distributions

5.-6. Structure of distributions

7. Distributions having support on compact sets or on subspaces

8. Tensor product of distributions

9. Product of distributions by functions

Applications to differential equations

10. Convolutions of distributions

11. Convolutions of distributions with smooth functions

12. The spaces Kr{Mp}, {DLr} and the structure of their generalized functions

13. Convolution equations

14. The spaces (S) and (S')

15. Fourier transforms of distributions

Problems

4 Convolutions and Fourier Transforms of Generalized Functions

1. Fourier transforms of fundamental functions

2. Fourier transforms of generalized functions

3. Convolutions of generalized functions

4. The convolution theorems

Problems

5 W Spaces

1. Theorems on complex analytic functions

2. Definition of W spaces

3. Operators in W spaces

4. Fourier transforms of W spaces

5. Nontriviality and richness of W spaces

Problems

6 Fourier Transforms of Entire Functions

1. Entire functions of order equal or less than p and of fast decrease

2. Entire functions of order equal or less than 1

3. Entire functions of order equal or less than 1 and of slow increase

4. Entire functions of order equal or less than p and of slow increase

5. Entire functions of order equal or less than p and of mildly fast increase

6. Entire functions of order equal or less than p and of fast increase

7. Proof of Lemma 2

Problems

7 The Cauchy Problem for Systems of Partial Differential Equations

1. Systems of partial differential equations and the Cauchy problem

2. Auxiliary theorems on functions of matrices

3. Uniqueness of solutions of the Cauchy problem

4. Existence of generalized solutions

5. Lemmas on convolutions

6. Existence theorems for parabolic systems

7. An existence theorem for hyperbolic systems

8. Existence theorems for correctly posed systems

9. Existence theorems for mildly incorrectly posed systems

10. An existence theorem for incorrectly posed systems

11. Nonhomogeneous systems with time-dependent coefficients

12. Systems of convolution equations

13. Difference-differential equations

14. Inverse theorems

15. Proof of the Seidenberg-Tarski theorem

Problems

8 The Cauchy Problem in Several Time Variables

1. Uniqueness and existence of generalized solutions

2. Sobolev's lemma

3. Proof of Theorem

4. Existence of classical solutions

5. The Goursat problem

Problems

9 S Spaces

1. Definition of S spaces

2. Operators in S spaces

3. Fourier transforms of S spaces

4. Nontriviality and richness of S spaces

Problems

10 Further Applications to Partial Differential Equations

1. A Phragmén-Lindelöf type theorem

2. A Liouville type theorem

3. Fundamental solutions of equations with constant coefficients

4. Special distributions and Radon's problem

5. Fundamental solutions for hyperbolic equations

Problems

11 Differentiability of Solutions of Partial Differential Equations

1. Hypoelliptic equations and their fundamental solutions

2.-3. Conditions for hypoellipticity

4. Examples of hypoelliptic equations

5. Nonhomogeneous equations

Problems

Bibliographical Remarks

Bibliography

Index for Spaces

Index

## Reading Group Guide

1 Linear Topological Spaces

1. Topological spaces and metric spaces

2. Linear topological spaces

3. Countably normed spaces

4. Continuous linear functionals

5. Weak and strong topologies

6. Perfect spaces

7. Linear operators

8. Inductive limits and unions of topological spaces

Problems

2 Spaces of Generalized Functions

1. Fundamental spaces and generalized functions

2. The spaces K{Mp}

3. The spaces Z{Mp}

4. Multiplication and the derivatives of generalized functions

5. Structure of generalized functions on K{Mp}

Problems

3 Theory of Distributions

1. Spaces of functions

2. Partition of unity

3. Definition and some properties of distributions

4. Derivatives of distributions

5.-6. Structure of distributions

7. Distributions having support on compact sets or on subspaces

8. Tensor product of distributions

9. Product of distributions by functions

Applications to differential equations

10. Convolutions of distributions

11. Convolutions of distributions with smooth functions

12. The spaces Kr{Mp}, {DLr} and the structure of their generalized functions

13. Convolution equations

14. The spaces (S) and (S')

15. Fourier transforms of distributions

Problems

4 Convolutions and Fourier Transforms of Generalized Functions

1. Fourier transforms of fundamental functions

2. Fourier transforms of generalized functions

3. Convolutions of generalized functions

4. The convolution theorems

Problems

5 W Spaces

1. Theorems on complex analytic functions

2. Definition of W spaces

3. Operators in W spaces

4. Fourier transforms of W spaces

5. Nontriviality and richness of W spaces

Problems

6 Fourier Transforms of Entire Functions

1. Entire functions of order equal or less than p and of fast decrease

2. Entire functions of order equal or less than 1

3. Entire functions of order equal or less than 1 and of slow increase

4. Entire functions of order equal or less than p and of slow increase

5. Entire functions of order equal or less than p and of mildly fast increase

6. Entire functions of order equal or less than p and of fast increase

7. Proof of Lemma 2

Problems

7 The Cauchy Problem for Systems of Partial Differential Equations

1. Systems of partial differential equations and the Cauchy problem

2. Auxiliary theorems on functions of matrices

3. Uniqueness of solutions of the Cauchy problem

4. Existence of generalized solutions

5. Lemmas on convolutions

6. Existence theorems for parabolic systems

7. An existence theorem for hyperbolic systems

8. Existence theorems for correctly posed systems

9. Existence theorems for mildly incorrectly posed systems

10. An existence theorem for incorrectly posed systems

11. Nonhomogeneous systems with time-dependent coefficients

12. Systems of convolution equations

13. Difference-differential equations

14. Inverse theorems

15. Proof of the Seidenberg-Tarski theorem

Problems

8 The Cauchy Problem in Several Time Variables

1. Uniqueness and existence of generalized solutions

2. Sobolev's lemma

3. Proof of Theorem

4. Existence of classical solutions

5. The Goursat problem

Problems

9 S Spaces

1. Definition of S spaces

2. Operators in S spaces

3. Fourier transforms of S spaces

4. Nontriviality and richness of S spaces

Problems

10 Further Applications to Partial Differential Equations

1. A Phragmén-Lindelöf type theorem

2. A Liouville type theorem

3. Fundamental solutions of equations with constant coefficients

4. Special distributions and Radon's problem

5. Fundamental solutions for hyperbolic equations

Problems

11 Differentiability of Solutions of Partial Differential Equations

1. Hypoelliptic equations and their fundamental solutions

2.-3. Conditions for hypoellipticity

4. Examples of hypoelliptic equations

5. Nonhomogeneous equations

Problems

Bibliographical Remarks

Bibliography

Index for Spaces

Index

## Interviews

1 Linear Topological Spaces

1. Topological spaces and metric spaces

2. Linear topological spaces

3. Countably normed spaces

4. Continuous linear functionals

5. Weak and strong topologies

6. Perfect spaces

7. Linear operators

8. Inductive limits and unions of topological spaces

Problems

2 Spaces of Generalized Functions

1. Fundamental spaces and generalized functions

2. The spaces K{Mp}

3. The spaces Z{Mp}

4. Multiplication and the derivatives of generalized functions

5. Structure of generalized functions on K{Mp}

Problems

3 Theory of Distributions

1. Spaces of functions

2. Partition of unity

3. Definition and some properties of distributions

4. Derivatives of distributions

5.-6. Structure of distributions

7. Distributions having support on compact sets or on subspaces

8. Tensor product of distributions

9. Product of distributions by functions

Applications to differential equations

10. Convolutions of distributions

11. Convolutions of distributions with smooth functions

12. The spaces Kr{Mp}, {DLr} and the structure of their generalized functions

13. Convolution equations

14. The spaces (S) and (S')

15. Fourier transforms of distributions

Problems

4 Convolutions and Fourier Transforms of Generalized Functions

1. Fourier transforms of fundamental functions

2. Fourier transforms of generalized functions

3. Convolutions of generalized functions

4. The convolution theorems

Problems

5 W Spaces

1. Theorems on complex analytic functions

2. Definition of W spaces

3. Operators in W spaces

4. Fourier transforms of W spaces

5. Nontriviality and richness of W spaces

Problems

6 Fourier Transforms of Entire Functions

1. Entire functions of order equal or less than p and of fast decrease

2. Entire functions of order equal or less than 1

3. Entire functions of order equal or less than 1 and of slow increase

4. Entire functions of order equal or less than p and of slow increase

5. Entire functions of order equal or less than p and of mildly fast increase

6. Entire functions of order equal or less than p and of fast increase

7. Proof of Lemma 2

Problems

7 The Cauchy Problem for Systems of Partial Differential Equations

1. Systems of partial differential equations and the Cauchy problem

2. Auxiliary theorems on functions of matrices

3. Uniqueness of solutions of the Cauchy problem

4. Existence of generalized solutions

5. Lemmas on convolutions

6. Existence theorems for parabolic systems

7. An existence theorem for hyperbolic systems

8. Existence theorems for correctly posed systems

9. Existence theorems for mildly incorrectly posed systems

10. An existence theorem for incorrectly posed systems

11. Nonhomogeneous systems with time-dependent coefficients

12. Systems of convolution equations

13. Difference-differential equations

14. Inverse theorems

15. Proof of the Seidenberg-Tarski theorem

Problems

8 The Cauchy Problem in Several Time Variables

1. Uniqueness and existence of generalized solutions

2. Sobolev's lemma

3. Proof of Theorem

4. Existence of classical solutions

5. The Goursat problem

Problems

9 S Spaces

1. Definition of S spaces

2. Operators in S spaces

3. Fourier transforms of S spaces

4. Nontriviality and richness of S spaces

Problems

10 Further Applications to Partial Differential Equations

1. A Phragmén-Lindelöf type theorem

2. A Liouville type theorem

3. Fundamental solutions of equations with constant coefficients

4. Special distributions and Radon's problem

5. Fundamental solutions for hyperbolic equations

Problems

11 Differentiability of Solutions of Partial Differential Equations

1. Hypoelliptic equations and their fundamental solutions

2.-3. Conditions for hypoellipticity

4. Examples of hypoelliptic equations

5. Nonhomogeneous equations

Problems

Bibliographical Remarks

Bibliography

Index for Spaces

Index

## Recipe

1. Topological spaces and metric spaces

2. Linear topological spaces

3. Countably normed spaces

4. Continuous linear functionals

5. Weak and strong topologies

6. Perfect spaces

7. Linear operators

8. Inductive limits and unions of topological spaces

Problems

2 Spaces of Generalized Functions

1. Fundamental spaces and generalized functions

2. The spaces K{Mp}

3. The spaces Z{Mp}

4. Multiplication and the derivatives of generalized functions

5. Structure of generalized functions on K{Mp}

Problems

3 Theory of Distributions

1. Spaces of functions

2. Partition of unity

3. Definition and some properties of distributions

4. Derivatives of distributions

5.-6. Structure of distributions

7. Distributions having support on compact sets or on subspaces

8. Tensor product of distributions

9. Product of distributions by functions

Applications to differential equations

10. Convolutions of distributions

11. Convolutions of distributions with smooth functions

12. The spaces Kr{Mp}, {DLr} and the structure of their generalized functions

13. Convolution equations

14. The spaces (S) and (S')

15. Fourier transforms of distributions

Problems

4 Convolutions and Fourier Transforms of Generalized Functions

1. Fourier transforms of fundamental functions

2. Fourier transforms of generalized functions

3. Convolutions of generalized functions

4. The convolution theorems

Problems

5 W Spaces

1. Theorems on complex analytic functions

2. Definition of W spaces

3. Operators in W spaces

4. Fourier transforms of W spaces

5. Nontriviality and richness of W spaces

Problems

6 Fourier Transforms of Entire Functions

1. Entire functions of order equal or less than p and of fast decrease

2. Entire functions of order equal or less than 1

3. Entire functions of order equal or less than 1 and of slow increase

4. Entire functions of order equal or less than p and of slow increase

5. Entire functions of order equal or less than p and of mildly fast increase

6. Entire functions of order equal or less than p and of fast increase

7. Proof of Lemma 2

Problems

7 The Cauchy Problem for Systems of Partial Differential Equations

1. Systems of partial differential equations and the Cauchy problem

2. Auxiliary theorems on functions of matrices

3. Uniqueness of solutions of the Cauchy problem

4. Existence of generalized solutions

5. Lemmas on convolutions

6. Existence theorems for parabolic systems

7. An existence theorem for hyperbolic systems

8. Existence theorems for correctly posed systems

9. Existence theorems for mildly incorrectly posed systems

10. An existence theorem for incorrectly posed systems

11. Nonhomogeneous systems with time-dependent coefficients

12. Systems of convolution equations

13. Difference-differential equations

14. Inverse theorems

15. Proof of the Seidenberg-Tarski theorem

Problems

8 The Cauchy Problem in Several Time Variables

1. Uniqueness and existence of generalized solutions

2. Sobolev's lemma

3. Proof of Theorem

4. Existence of classical solutions

5. The Goursat problem

Problems

9 S Spaces

1. Definition of S spaces

2. Operators in S spaces

3. Fourier transforms of S spaces

4. Nontriviality and richness of S spaces

Problems

10 Further Applications to Partial Differential Equations

1. A Phragmén-Lindelöf type theorem

2. A Liouville type theorem

3. Fundamental solutions of equations with constant coefficients

4. Special distributions and Radon's problem

5. Fundamental solutions for hyperbolic equations

Problems

11 Differentiability of Solutions of Partial Differential Equations

1. Hypoelliptic equations and their fundamental solutions

2.-3. Conditions for hypoellipticity

4. Examples of hypoelliptic equations

5. Nonhomogeneous equations

Problems

Bibliographical Remarks

Bibliography

Index for Spaces

Index