Methods of the theory of generalized functions / V.S. Vladimirov.

Format:

Book

Author/Creator:

Vladimirov, V. S. (Vasiliĭ Sergeevich)

Contributor:

Alumni and Friends Memorial Book Fund.

Series:

Analytical methods and special functions ; v. 6.

Analytical methods and special functions ; v. 6

Language:

English

Subjects (All):

Theory of distributions (Functional analysis).

Integral transforms.

Mathematical physics.

Physical Description:

xiv, 311 pages : illustrations ; 26 cm.

Place of Publication:

London ; New York : Taylor & Francis, 2002.

Contents:

Chapter 1. Generalized Functions and their Properties 5

1.2. The space of test functions D(O) 6

1.3. The space of generalized functions D'(O) 10

1.4. The completeness of the space of generalized functions D'(O) 12

1.5. The support of a generalized function 13

1.6. Regular generalized functions 15

1.7. Measures 16

1.8. Sochozki formulae 19

1.9. Change of variables in generalized functions 21

1.10. Multiplication of generalized functions 23

2. Differentiation of Generalized Functions 25

2.1. Derivatives of generalized functions 25

2.2. The antiderivative (primitive) of a generalized function 27

2.4. The local structure of generalized functions 35

2.5. Generalized functions with compact support 36

2.6. Generalized functions with point support 37

2.7. Generalized functions P([pi subscript v vertical bar]x[vertical bar superscript alpha-1]) 39

3. Direct Product of Generalized Functions 41

3.2. The properties of a direct product 43

3.4. Generalized functions that are smooth with respect to some of the variables 48

4. The Convolution of Generalized Functions 50

4.2. The properties of a convolution 53

4.3. The existence of a convolution 57

4.4. Cones in R[superscript n] 59

4.5. Convolution algebras D'([Gamma]+) and D'([Gamma]) 63

4.6. Mean functions of generalized functions 64

4.7. Multiplication of generalized functions 66

4.8. Convolution as a continuous linear translation-invariant operator 66

5. Tempered Generalized Functions 74

5.1. The space S of test (rapidly decreasing) functions 74

5.2. The space S' of tempered generalized functions 77

5.3. Examples of tempered generalized functions and elementary operations in S' 78

5.4. The structure of tempered generalized functions 80

5.5. The direct product of tempered generalized functions 81

5.6. The convolution of tempered generalized functions 82

5.7. Homogeneous generalized functions 85

Chapter 2. Integral Transformations of Generalized Functions 89

6. The Fourier Transform of Tempered Generalized Functions 89

6.1. The Fourier transform of test functions in S 89

6.2. The Fourier transform of tempered generalized functions 90

6.3. Properties of the Fourier transform 92

6.4. The Fourier transform of generalized functions with compact support 93

6.5. The Fourier transform of a convolution 94

6.7. The Mellin transform 109

7. Fourier Series of Periodic Generalized Functions 113

7.1. The definition and elementary properties of periodic generalized functions 113

7.2. Fourier series of periodic generalized functions 116

7.3. The convolution algebra D'[subscript T] 117

8. Positive Definite Generalized Functions 121

8.1. The definition and elementary properties of positive definite generalized functions 121

8.2. The Bochner-Schwartz theorem 123

9. The Laplace Transform of Tempered Generalized Functions 126

9.1. Definition of the Laplace transform 126

9.2. Properties of the Laplace transform 128

10. The Cauchy Kernel and the Transforms of Cauchy-Bochner and Hilbert 133

10.1. The space H[subscript s] 133

10.2. The Cauchy kernel K[subscript C](z) 138

10.3. The Cauchy-Bochner transform 144

10.4. The Hilbert transform 146

10.5. Holomorphic functions of the class H[superscript (s) subscript a] (C) 147

10.6. The generalized Cauchy-Bochner representation 151

11. Poisson Kernel and Poisson Transform 152

11.1. The definition and properties of the Poisson kernel 152

11.2. The Poisson transform and Poisson representation 155

11.3. Boundary values of the Poisson integral 157

12. Algebras of Holomorphic Functions 159

12.1. The definition of the H[subscript +] (C) and H (C) algebras 160

12.2. Isomorphism of the algebras S'(C* +) - H[subscript +] (C) and S'(C*) - H (C) 160

12.3. The Paley-Wiener-Schwartz theorem and its generalizations 165

12.4. The space H[subscript a](C) is the projective limit of the spaces H[subscript a'](C') 166

12.5. The Schwartz representation 168

12.6. A generalization of the Phragmen-Lindelof theorem 171

13. Equations in Convolution Algebras 171

13.1. Divisors of unity in the H[subscript +] (C) and H(C) algebras 171

13.2. On division by a polynomial in the H(C) algebra 172

13.3. Estimates for holomorphic functions with nonnegative imaginary part in T[subscript C] 174

13.4. Divisors of unity in the algebra W(C) 177

14. Tauberian Theorems for Generalized Functions 179

14.1. Preliminary results 179

14.2. General Tauberian theorem 183

14.3. One-dimensional Tauberian theorems 186

14.4. Tauberian and Abelian theorems for nonnegative measures 187

14.5. Tauberian theorems for holomorphic functions of bounded argument 188

Chapter 3. Some Applications in Mathematical Physics 191

15. Differential Operators with Constant Coefficients 191

15.1. Fundamental solutions in D' 191

15.2. Tempered fundamental solutions 194

15.3. A descent method 196

15.5. A comparison of differential operators 207

15.6. Elliptic and hypoelliptic operators 210

15.7. Hyperbolic operators 212

15.8. The sweeping principle 212

16. The Cauchy Problem 213

16.1. The generalized Cauchy problem for a hyperbolic equation 213

16.2. Wave potential 216

16.3. Surface wave potentials 220

16.4. The Cauchy problem for the wave equation 222

16.5. A statement of the generalized Cauchy problem for the heat equation 224

16.6. Heat potential 224

16.7. Solution of the Cauchy problem for the heat equation 228

17. Holomorphic Functions with Nonnegative Imaginary Part in T[subscript C] 229

17.2. Properties of functions of the class P[subscript +](T[superscript C]) 231

17.3. Estimates of the growth of functions of the class H[subscript +](T[superscript C]) 238

17.4. Smoothness of the spectral function 240

17.5. Indicator of growth of functions of the class P[subscript +](T[superscript C]) 242

17.6. An integral representation of functions of the class H[subscript +](T[superscript C]) 245

18. Holomorphic Functions with Nonnegative Imaginary Part in T[superscript n] 249

18.1. Lemmas 249

18.2. Functions of the classes H[subscript +](T[superscript 1]) and P[subscript +](T[superscript 1]) 254

18.3. Functions of the class P[subscript +](T[superscript n]) 258

18.4. Functions of the class H[subscript +](T[superscript n]) 263

19. Positive Real Matrix Functions in T[superscript C] 266

19.1. Positive real functions in T[superscript C] 267

19.2. Positive real matrix functions in T[superscript C] 269

20. Linear Passive Systems 271

20.2. Corollaries to the condition of passivity 273

20.3. The necessary and sufficient conditions for passivity 277

20.4. Multidimensional dispersion relations 282

20.5. The fundamental solution and the Cauchy problem 285

20.6. What differential and difference operators are passive operators? 287

20.8. Quasiasymptotics of the solutions of equations in convolutions 294

21. Abstract Scattering Operator 295

21.1. The definition and properties of an abstract scattering matrix 295

21.2. A description of abstract scattering matrices 298

21.3. The relationship between passive operators and scattering operators 299.

Notes:

Includes bibliographical references (pages 303-308) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0415273560

OCLC:

50824779