Generalized Functions and Partial

This self-contained treatment develops the theory of generalized functions and the theory of distributions, and it systematically applies them to a variety of problems in partial differential equations. It covers Cauchy and Goursat problems, fundamental solutions, existence and differentiality of solutions of equations with constants, coefficients, and related topics. 1963 edition

CHAPTER 1 LINEAR TOPOLOGICAL SPACES,

1

(24)

1. Topological spaces and metric spaces,

1

(2)

2. Linear topological spaces,

3

(3)

3. Countably normed spaces,

6

(3)

4. Continuous linear functionals,

9

(2)

5. Weak and strong topologies,

11

(4)

6. Perfect spaces,

15

(3)

7. Linear operators,

18

(2)

8. Inductive limits and unions of topological spaces,

20

(4)

Problems,

24

(1)

CHAPTER 2 SPACES OF GENERALIZED FUNCTIONS,

25

(18)

1. Fundamental spaces and generalized functions,

25

(4)

2. The spaces K {M,p},

29

(4)

3. The spaces Z {Mp},

33

(1)

4. Multiplication and the derivatives of generalized functions,

34

(3)

5. Structure of generalized functions on K{Mp},

37

(4)

Problems,

41

(2)

CHAPTER 3 THEORY OF DISTRIBUTIONS,

43

(58)

1. Spaces of functions,

43

(2)

2. Partition of unity,

45

(2)

3. Definition and some properties of distributions,

47

(5)

4. Derivatives of distributions,

52

(7)

5. Structure of distributions,

59

(4)

6. Structure of distributions (continued),

63

(5)

7. Distributions having support on compact sets or on subspaces,

68

(2)

8. Tensor product of distributions,

70

(3)

9. Product of distributions by functions. Applications to differential equations,

73

(3)

10. Convolutions of distributions,

76

(3)

11. Convolutions of distributions with smooth functions,

79

(4)

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

83

(5)

13. Convolution equations,

88

(2)

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

90

(3)

15. Fourier transforms of distributions,

93

(5)

Problems,

98

(3)

CHAPTER 4 CONVOLUTIONS AND FOURIER TRANSFORMS OF GENERALIZED FUNCTIONS,

101

(12)

1. Fourier transforms of fundamental functions,

101

(2)

2. Fourier transforms of generalized functions,

103

(1)

3. Convolutions of generalized functions,

104

(6)

4. The convolution theorems,

110

(2)

Problems,

112

(1)

CHAPTER 5 W SPACES,

113

(27)

1. Theorems on complex analytic functions,

113

(12)

2. Definition of W spaces,

125

(3)

3. Operators in W spaces,

128

(3)

4. Fourier transforms of W spaces,

131

(5)

5. Nontriviality and richness of W spaces,

136

(3)

Problems,

139

(1)

CHAPTER 6 FOURIER TRANSFORMS OF ENTIRE FUNCTIONS,

140

(24)

1. Entire functions of order less than or equal to p and of fast decrease, 140

2. Entire functions of order less than or equal to 1, 142

3. Entire functions of order less than or equal to 1 and of slow increase, 145

4. Entire functions of order less than or equal to p and of slow increase, 147

5. Entire functions of order less than or equal to p and of mildly fast increase, 150

6. Entire functions of order less than or equal to p and of fast increase, 158

7. Proof of Lemma 2,

159

(4)

Problems,

163

(1)

CHAPTER 7 THE CAUCHY PROBLEM FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS,

164

(73)

1. Systems of partial differential equations and the Cauchy problem,

164

(4)

2. Auxiliary theorems on functions of matrices,

168

(9)

3. Uniqueness of solutions of the Cauchy problem,

177

(7)

4. Existence of generalized solutions,

184

(2)

5. Lemmas on convolutions,

186

(5)

6. Existence theorems for parabolic systems,

191

(5)

7. An existence theorem for hyperbolic systems,

196

(2)

8. Existence theorems for correctly posed systems,

198

(2)

9. Existence theorems for mildly incorrectly posed systems,

200

(5)

10. An existence theorem for incorrectly posed systems,

205

(1)

11. Nonhomogeneous systems with time-dependent coefficients,

206

(2)

12. Systems of convolution equations,

208

(2)

13. Difference-differential equations,

210

(8)

14. Inverse theorems,

218

(7)

15. Proof of the Seidenberg-Tarski theorem,

225

(10)

Problems,

235

(2)

CHAPTER 8 THE CAUCHY PROBLEM IN SEVERAL TIME VARIABLES,

237

(21)

1. Uniqueness and existence of generalized solutions,

237

(4)

2. Sobolev's lemma,

241

(2)

3. Proof of Theorem 2,

243

(5)

4. Existence of classical solutions,

248

(2)

5. The Goursat problem,

250

(7)

Problems,

257

(1)

CHAPTER 9 S SPACES,

258

(15)

1. Definition of S spaces,

258

(6)

2. Operators in S spaces,

264

(2)

3. Fourier transforms of S spaces,

266

(4)

4. Nontriviality and richness of S spaces,

270

(2)

Problems,

272

(1)

CHAPTER 10 FURTHER APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS,

273

(28)

1. A Phragmen-Lindelof type theorem,

273

(6)

2. A Liouville type theorem,

279

(6)

3. Fundamental solutions of equations with constant coefficients,

285

(4)

4. Special distributions and Radon's problem,

289

(6)

5. Fundamental solutions for hyperbolic equations,

295

(4)

Problems,

299

(2)

CHAPTER 11 DIFFERENTIABILITY OF SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS,

301

(25)

1. Hypoelliptic equations and their fundamental solutions,

301

(3)

2. Conditions for hypoellipticity,

304

(9)

3. Conditions for hypoellipticity (continued),

313

(3)

4. Examples of hypoelliptic equations,

316

(4)

5. Nonhomogeneous equations,

320

(5)

Problems,

325

(1)

BIBLIOGRAPHICAL REMARKS,

326

(3)

BIBLIOGRAPHY,

329

(4)

INDEX FOR SPACES,

333

(2)

INDEX,

335

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Generalized Functions and Partial Differential Equations

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