Methods of the Theory of Generalized Functions

Cover......Page 1
Sub Title......Page 2
List of Publications......Page 3
Title......Page 4
ISBN 0415273560 ......Page 5
Contents......Page 6
Preface......Page 12
Symbols and Definitions......Page 16
1.1. Introduction. ......Page 20
1.2. The space of test functions D(O). ......Page 21
1.3. The space of generalized functions D\'\'(O). ......Page 25
1.4. The completeness of the space of generalized functions D\' (0). ......Page 27
1.5. The support of a generalized function. ......Page 28
1.6. Regular generalized functions. ......Page 30
1. 7. Measures. ......Page 31
1.8. Sochozki formulae. ......Page 34
1.9. Change of variables in generalized functions. ......Page 36
1.10. Multiplication of generalized functions. ......Page 38
2.1. Derivatives of generalized functions. ......Page 40
2.2. The antiderivative (primitive) of a generalized function. ......Page 42
2.3. Examples. ......Page 44
2.4. The local structure of generalized functions. ......Page 50
2.5. Generalized functions with compact support. ......Page 51
2.6. Generalized functions with point support. ......Page 52
2.7. Generalized functions ......Page 54
3.1. The definition of a direct product. ......Page 56
3.2. The properties of a direct product. ......Page 58
3.3. Some applications. ......Page 61
3.4. Generalized functions that are smooth with respect to some of the variables......Page 63
4.1. The definition of convolution. ......Page 65
4.2. The properties of a convolution. ......Page 68
4.3. The existence of a convolution. ......Page 72
4.4. Cones in R^n......Page 74
4.5. Convolution algebras D\'(Gamma+) and D\'(Gamma). ......Page 78
4.6. Mean functions of generalized functions. ......Page 79
4.8. Convolution as a continuous linear translation-invariant operator......Page 81
4.9. Some applications. ......Page 83
5.1. The space S of test (rapidly decreasing) functions. ......Page 89
5.2. The space S\' of tempered generalized functions. ......Page 92
5.3. Examples of tempered generalized functions and elementary operations in S\'......Page 93
5.4. The structure of tempered generalized functions. ......Page 95
5.5. The direct product of tempered generalized functions. ......Page 96
5.6. The convolution of tempered generalized functions. ......Page 97
5.7. Homogeneous generalized functions. ......Page 100
6.1. The Fourier transform of test functions in S. ......Page 104
6.2. The Fourier transform of tempered generalized functions. ......Page 105
6.3. Properties of the Fourier transform. ......Page 107
6.4. The Fourier transform of generalized functions with compact ......Page 108
6.5. The Fourier transform of a convolution. ......Page 109
6.6. Examples. ......Page 111
6.7. The Mellin transform. ......Page 124
7.1. The definition and elementary properties of periodic generalized functions. ......Page 128
7.2. Fourier series of periodic generalized functions. ......Page 131
7.3. The convolution algebra D\'_T......Page 132
7.4. Examples. ......Page 134
8.1. The definition and elementary properties of positive definite generalized functions......Page 136
8.2. The Bochner-Schwartz theorem. ......Page 138
8,3. Examples. ......Page 140
9.1. Definition of the Laplace transform. ......Page 141
9.2. Properties of the Laplace transform, ......Page 143
9.3. Examples. ......Page 145
10.1. The space Hs. ......Page 148
10.2. The Cauchy kernel Kc(z). ......Page 153
10.3. The Cauchy-Bochner transform. ......Page 159
10.4. The Hilbert transform. ......Page 161
10.5. HoloIllorphic functions of the class Hs(C)......Page 162
10.6. The generalized Cauchy-Bochner representation. ......Page 166
11.1. The definition and properties of the Poisson kernel. ......Page 167
11.2. The Poisson transform and Poisson representation. ......Page 170
11.3. Boundary values of the Poisson integral. ......Page 172
12. Algebras of Holomorphic Functions ......Page 174
12.2. Isomorphism of the algebras S\'(C*+) ......Page 175
12.3. The Paley-Wiener-Schwartz theorem and its generalizations. ......Page 180
12.4. The space Ha(C) is the projective limit of the spaces Ha,(C / ). ......Page 181
12.5. The Schwartz representation. ......Page 183
13.1. Divisors of unity in the H+(C) and H(C) algebras. ......Page 186
13.2. On division by a polynomial in the H(C) algebra. ......Page 187
13.3. Estimates for holomorphic functions with nonnegative imaginary part in T C......Page 189
13.5. Example. ......Page 192
14.1. Preliminary results. ......Page 194
14.2. General Tauberian theorem. ......Page 198
14.3. One-dimensional Tauberian theorems. ......Page 201
14.4. Tauberian and Abelian theorems for nonnegative measures. ......Page 202
14.5. Tauberian theorems for holomorphic functions of bounded argument......Page 203
15.1. Fundamental solutions in D\'. ......Page 206
15.2. Tempered fundamental solutions. ......Page 209
15.3. A descent method. ......Page 211
15.4. Examples. ......Page 214
15.5. A comparison of differential operators. ......Page 222
15.6. Elliptic and hypoelliptic operators. ......Page 225
15.8. The sweeping principle. ......Page 227
16.1. The generalized Cauchy problem for a hyperbolic equation. ......Page 228
16.2. Wave potential. ......Page 231
16.3. Surface wave potentials. ......Page 235
16.4. The Cauchy problem for the wave equation. ......Page 237
16.6. Heat potential. ......Page 239
16.7. Solution of the Cauchy problem for the heat equation. ......Page 243
17.1. Preliminary remarks. ......Page 244
17.2. Properties of functions of the class P+(T C ). ......Page 246
17.3. Estimates of the growth of functions of the class H+(T c ). ......Page 253
17.4. Smoothness of the spectral function. ......Page 255
17.6. An integral representation of functions of the class H + (T C ). ......Page 260
18.1. Lemmas. ......Page 264
18.2. Functions of the classes H+(T 1 ) and P+(T l ). ......Page 269
18.3. Functions of the class P + (Tn). ......Page 273
18.4. Functions of the class H+(T n ). ......Page 278
19. Positive Real Matrix Functions in T^ C ......Page 281
19.1. Positive real functions in TC. ......Page 282
19.2. Positive real matrix functions in T C ......Page 284
20.1. Introduction. ......Page 286
20.2. Corollaries to the condition of passivity. ......Page 288
20.3. The necessary and sufficient conditions for passivity. ......Page 292
20.4. Multidimensional dispersion relations. ......Page 297
20.5. The fundamental solution and the Cauchy problem. ......Page 300
20.6. What differential and difference operators are passive operators? ......Page 302
20.7. Examples. ......Page 305
20.8. Quasiasymptotics of the solutions of systems of equations in convolutions. ......Page 309
21.1. The definition and properties of an abstract scattering matrix. ......Page 310
21.2. A description of abstract scattering matrices. ......Page 313
21.3. The relationship between passive operators and scattering operators. ......Page 314
BIBLIOGRAPHY ......Page 318
Index ......Page 324
Back Cover......Page 327