The theory of generalised functions

Title......Page 1
ISBN......Page 2
Contents......Page 5
Preface......Page 9
1.1 Preliminary definitions......Page 13
1.2 Sequences......Page 15
1.3 Functions......Page 17
1.4 Topological space......Page 19
1.5 Compactness......Page 21
1.6 Metric spaces......Page 24
1.7 Function spaces......Page 27
1.8 Numbers......Page 31
1.9 Limits......Page 33
1.10 Series......Page 37
1.11 Derivatives......Page 40
1.12 Integration......Page 43
2.1 Good functions......Page 56
2.2 Fourier transforms......Page 58
3.1 Generalised functions......Page 65
3.2 Conventional functions......Page 70
3.3 The derivative......Page 75
3.4 The Fourier transform......Page 83
3.5 Limits as generalised functions......Page 87
3.6 Classification of generalised functions......Page 93
4.2 Powers of x......Page 106
4.3 Even and odd generalised functions......Page 109
4.4 Singular integrals......Page 111
4.5 Fourier transforms......Page 121
4.6 Generalised functions containing a logarithm......Page 127
4.7 Integration......Page 133
5.1 General properties......Page 140
5.2 Hermite polynomials......Page 144
5.3 The expansion of a generalised function in Hermite polynomials......Page 149
5.4 Functions of the class L2......Page 153
5.5 Fourier series......Page 156
5.6 Periodic generalised functions......Page 161
5.7 Poisson summation formula......Page 167
An alternative definition of a periodic generalised function......Page 172
6.1 Multiplication......Page 174
6.2 Division......Page 184
6.3 The convolution......Page 188
6.4 Properties of the convolution......Page 192
6.5 A generalisation of the convolution......Page 195
6.6 The convolution for the class K+......Page 212
7 Several variables......Page 222
7.1 Good functions......Page 223
7.2 Generalised functions......Page 228
7.3 The derivative......Page 232
7.4 The Fourier transform......Page 236
7.5 The general form of a generalised function......Page 240
7.6 The direct product......Page 247
7.7 Some special generalised functions......Page 254
7.8 Multiplication......Page 269
7.9 Integration with respect to a parameter......Page 275
8.1 Rotation of axes......Page 279
8.2 Change of variable: one-dimensional case......Page 285
8.3 Change of variables: several dimensions......Page 299
8.4 Delta functions......Page 304
8.5 Periodic and other changes of variables......Page 312
8.6 Singular integrals......Page 320
8.7 The hyperbolic and ultrahyperbolic distances......Page 322
8.8 Fourier transforms of the hyperbolic and ultrahyperbolic distances......Page 341
8.9 The general quadratic form......Page 349
8.10 The class K+......Page 352
9.1 The Riemann-Lebesgue lemma......Page 357
9.2 Generalised functions with a finite number of singularities in one dimension......Page 361
9.3 The method of stationary phase......Page 369
9.4 Generalised functions with isolated singularities......Page 374
9.5 Integrals over a finite domain......Page 376
9.6 Stationary phase in higher dimensions......Page 391
9.7 Inverse polynomials......Page 398
10.1 Integral equations......Page 402
10.2 Hubert transforms......Page 408
10.3 Ordinary linear differential equations......Page 417
10.4 Linear partial differential equations......Page 422
10.5 Approximate behaviour of elementary solutions......Page 439
10.6 Covariance and generalised functions of positive type......Page 441
11.1 Weak functions in one dimension......Page 446
11.2 The weak derivative and the weak limit......Page 449
11.3 The classification of weak functions......Page 451
11.4 Sundry results......Page 456
11.5 Multiplication......Page 459
11.6 The convolution......Page 463
11.7 Volterra integral equations......Page 470
11.8 Ordinary linear differential equations......Page 472
11.9 Linear differential equations with constant coefficients......Page 474
11.10 The operational method......Page 479
11.11 Weak functions of several variables......Page 490
11.12 Integration with respect to a parameter......Page 496
11.13 Change of variable: single variable case......Page 497
11.14 Change of variables: several variables......Page 499
11.15 Ultradistributions and Fourier transforms......Page 500
11.16 The relation between weak functions and distributions......Page 507
Appendix: Titchmarsh's theorem......Page 509
12.1 The Laplace transform......Page 512
12.2 Ordinary differential equations......Page 518
12.3 Integral equations......Page 521
12.4 Laplace transform of a weak function of several variables......Page 522
12.5 Partial differential equations......Page 525
12.6 The bilateral Laplace transform......Page 528
12.7 Integral equations......Page 531
12.8 The Wiener-Hopf integral equation......Page 532
12.9 A related partial differential equation......Page 536
Table of Fourier transforms......Page 541
Table of Laplace transforms......Page 545
Index......Page 546